Theorem opelopabga

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypothesis	opelopabga.1	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	opelopabga	$$⊢\left(A\in V\wedge B\in W\right)\to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		opelopabga.1	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		elopab	$$⊢⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \exists x\exists y\left(⟨A,B⟩=⟨x,y⟩\wedge \phi \right)$$
3	1	copsex2g	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\exists x\exists y\left(⟨A,B⟩=⟨x,y⟩\wedge \phi \right)\leftrightarrow \psi \right)$$
4	2 3	syl5bb	$$⊢\left(A\in V\wedge B\in W\right)\to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \psi \right)$$