How to break lines in a long equation in the following way?

Question

Just what I titled out, how to realize the following results?

Answer 1

I suggest you load the amsmath package and use its bmatrix ("bracketed matrix") environment.

\documentclass{article}
\usepackage{amsmath,bm,newtxtext,newtxmath}

\begin{document}
\[
(\forall \bm{p}) 
\begin{bmatrix}
(\forall x) (\varphi(x,\bm{p}) \leftrightarrow \cdots ) \to \\
\bigl((\varphi(0,\bm{p}) \wedge (\varphi(x,\bm{p}) \to
  \varphi(x+1,\bm{p}))) \to \cdots \bigr)
\end{bmatrix}
\]
\end{document}

Answer 2

You can use bmatrix from amsmath package for this. If you need also italic bold math, load the bm package and use \bm{..} instead of \mathbf{..}.

\documentclass[a4paper,10pt]{article}
\usepackage{amsmath}
\begin{document}

\[
(\forall \mathbf{p})
\begin{bmatrix}
(\forall x) ( \varphi(x,\mathbf{p}) \leftrightarrow \neg\psi(x,\mathbf{p}))\rightarrow\\
\bigl((\varphi(0,\mathbf{p}) \wedge (\varphi(x,\mathbf{p})\to\varphi(x+1,\mathbf{p})))\to(\forall x)\varphi(x,\mathbf{p})\bigr)
\end{bmatrix}
\]

\end{document}

Answer 3

In my opinion, gathered is the best choice. I also implemented new commands, because in this context the arrows should be binary operation symbols.

\documentclass{article}
\usepackage{newtxtext,newtxmath}
\usepackage{amsmath,bm}

\newcommand{\vp}{\bm{p}}
\newcommand{\liff}{\mathbin{\leftrightarrow}}
\newcommand{\limpl}{\mathbin{\rightarrow}}

\begin{document}

\[
(\forall \vp)
  \left[
  \begin{gathered}
  (\forall x) (\varphi(x,\vp) \liff \dotsb ) \limpl {}\\
  \bigl((\varphi(0,\vp) \land (\varphi(x,\vp) \limpl \varphi(x+1,\vp))) \to \dotsb \bigr)
  \end{gathered}
  \right]
\]

\end{document}