Theorem opelopaba

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

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

Proof

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