Theorem opelopab

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by NM, 16-May-1995)

	Ref	Expression
Hypotheses	opelopab.1	$$⊢A\in \mathrm{V}$$
	opelopab.2	$$⊢B\in \mathrm{V}$$
	opelopab.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	opelopab.4	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	opelopab	$$⊢⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \chi$$

Proof

Step	Hyp	Ref	Expression
1		opelopab.1	$$⊢A\in \mathrm{V}$$
2		opelopab.2	$$⊢B\in \mathrm{V}$$
3		opelopab.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
4		opelopab.4	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
5	3 4	opelopabg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \chi \right)$$
6	1 2 5	mp2an	$$⊢⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \chi$$