LaTeX \if condition has a good explanation of \if and a link to TeX by Topic explaining it a bit more already, so we can probably focus on this particular example here.
The test
\if!\@subheader!\else\noindent{\trfont{\@subheader}}\fi
is indeed intended to check if \@subheader is empty. It relies on the assumption that the contents of \@subheader will never start with !.
\documentclass{article}
\newcommand*{\mytest}{%
\if!\foo!
empty
\else
\emph{not} empty
\fi
}
\begin{document}
\def\foo{\empty}
\mytest
\def\foo{}
\mytest
\def\foo{Hello}
\mytest
\def\foo{!oohh, nasty}
\mytest
\def\foo{That works! Phew!}
\mytest
\def\foo{ }
\mytest
\end{document}
shows a few examples of the conditional in action.
\if expands the following content until it finds two unexpandable tokens. Those tokens are then compared.
The ! is already an unexpandable token, so the test compares ! to the first unexpandable token that occurs when \@subheader! is expanded.
Suppose \@subheader is empty in the sense that it expands to nothing (if it is fully expanded). This happens for example if we have \def\@subheader{} or even \def\@subheader{\@empty}, where \@subheader is defined to expand to \@empty, which in turn expands to nothing. Then \if!\@subheader! compares ! to ! since \@subheader expands to nothing and just vanishes. That test is true and so we get the <true> branch, which does nothing in the original code.
Suppose \@subheader is not empty and that its first token after full expansion is not !. If for example \@subheader is \def\@subheader{Hello}, then \if!\@subheader! compares ! with H and the rest of the expansion and the trailing ! (ello!) live on in the <true> branch. Since ! and H are different, we get the <false> branch, though and don't get to see the excess letters.
Suppose \@subheader expands to something which contains ! as the first token, e.g. \def\@subheader{!oohh, nasty} Then \if!\@subheader! compares ! to ! and enters the <true> branch. The expanded rest of the \@subheader (oohh, nasty) lives on there and produces unwanted additional output.
If there was no trailing ! in \if!\@subheader! TeX would have to keep looking for the second token in case \@subheader expands to nothing.
\@subheaderis empty (or\@empty)\ifcompares the two!s which yields false and means that nothing happens since the<true>branch is otherwise empty. If\@subheaderis not empty,\ifcompares!to the first 'thing' in\@subheader. The test relies on the assumption that this comparison is false. Then the\else/<false>branch is executed. You can confuse this test if you start your\@subheaderwith an!. Note that\ifexpands the stuff that follows it until it gets two unexpandable tokens (for example letters). – moewe Mar 21 '19 at 08:42\@subheaderneeded? Is it correct that it comapares the character "!" with the characters found in\@subheader? In the link above I read that everything after\ifwill be expanded until you get two unexpandable tokens. Would you please explain what happens if\@subheaderis set to the word hello, what would happen then? – JezuzStardust Mar 21 '19 at 09:48\ifx\@empty\@subheader\relax– John Kormylo Mar 21 '19 at 15:11