Theorem opelopabaf

Description: The law of concretion. Theorem 9.5 of Quine p. 61. This version of opelopab uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2013) (Proof shortened by Mario Carneiro, 18-Nov-2016)

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

Proof

Step	Hyp	Ref	Expression
1		opelopabaf.x	$$⊢Ⅎx\psi$$
2		opelopabaf.y	$$⊢Ⅎy\psi$$
3		opelopabaf.1	$$⊢A\in \mathrm{V}$$
4		opelopabaf.2	$$⊢B\in \mathrm{V}$$
5		opelopabaf.3	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
6		opelopabsb	$$⊢⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi$$
7		nfv	$$⊢ℲxB\in \mathrm{V}$$
8	1 2 7 5	sbc2iegf	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi \leftrightarrow \psi \right)$$
9	3 4 8	mp2an	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi \leftrightarrow \psi$$
10	6 9	bitri	$$⊢⟨A,B⟩\in \left\{⟨x,y⟩|\phi \right\}\leftrightarrow \psi$$