Starting from version 1.5 of NewTX, released on 2016/08/12, there is no alternate version of the quantifiers:
As of version 1.5, the old txfonts versions of \forall, \exists and \nexists have been removed and the Alt versions substituted. Both \forall and \forallAlt generate ∀, and similarly with \exists and \nexists.
Indeed, the following document shows what's claimed in the documentation.
\documentclass{article}
\usepackage{newtxtext,newtxmath}
\begin{document}
\verb|\exists\forall|: $\exists\forall$
\verb|\existsAlt\forallAlt|: $\existsAlt\forallAlt$
\texttt{\meaning\exists-\meaning\existsAlt}
\texttt{\meaning\forall-\meaning\forallAlt}
\end{document}

If I compile the same with TeX Live 2015, I get

The same document with px instead of tx produces what follows:

With TeX Live 2015, \existsAlt and \forallAlt produce errors. Removing them
\documentclass{article}
\usepackage{newpxtext,newpxmath}
\begin{document}
\verb|\exists\forall|: $\exists\forall$
\end{document}
produces

As you see, the ugly versions have been replaced in the current version of newpxmath.
newtxmathornewpxmath? – Jan 12 '17 at 19:54newtx: As of version 1.5, the oldtxfontsversions of\forall,\existsand\nexistshave been removed and theAltversions substituted. Both\foralland\forallAltgenerate ∀, and similarly with\existsand\nexists.” – egreg Jan 12 '17 at 22:25