Theorem copsex2g

Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995) Use a similar proof to copsex4g to reduce axiom usage. (Revised by SN, 1-Sep-2024)

	Ref	Expression
Hypothesis	copsex2g.1	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	copsex2g	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\exists x\exists y\left(⟨A,B⟩=⟨x,y⟩\wedge \phi \right)\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		copsex2g.1	$$⊢\left(x=A\wedge y=B\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		eqcom	$$⊢⟨A,B⟩=⟨x,y⟩\leftrightarrow ⟨x,y⟩=⟨A,B⟩$$
3		vex	$$⊢x\in \mathrm{V}$$
4		vex	$$⊢y\in \mathrm{V}$$
5	3 4	opth	$$⊢⟨x,y⟩=⟨A,B⟩\leftrightarrow \left(x=A\wedge y=B\right)$$
6	2 5	bitri	$$⊢⟨A,B⟩=⟨x,y⟩\leftrightarrow \left(x=A\wedge y=B\right)$$
7	6	anbi1i	$$⊢\left(⟨A,B⟩=⟨x,y⟩\wedge \phi \right)\leftrightarrow \left(\left(x=A\wedge y=B\right)\wedge \phi \right)$$
8	7	2exbii	$$⊢\exists x\exists y\left(⟨A,B⟩=⟨x,y⟩\wedge \phi \right)\leftrightarrow \exists x\exists y\left(\left(x=A\wedge y=B\right)\wedge \phi \right)$$
9		id	$$⊢\left(x=A\wedge y=B\right)\to \left(x=A\wedge y=B\right)$$
10	9 1	cgsex2g	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\exists x\exists y\left(\left(x=A\wedge y=B\right)\wedge \phi \right)\leftrightarrow \psi \right)$$
11	8 10	bitrid	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\exists x\exists y\left(⟨A,B⟩=⟨x,y⟩\wedge \phi \right)\leftrightarrow \psi \right)$$