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	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
Assertion	copsex2g	|- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		copsex2g.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
2		eqcom	|- ( <. A , B >. = <. x , y >. <-> <. x , y >. = <. A , B >. )
3		vex	|- x e. _V
4		vex	|- y e. _V
5	3 4	opth	|- ( <. x , y >. = <. A , B >. <-> ( x = A /\ y = B ) )
6	2 5	bitri	|- ( <. A , B >. = <. x , y >. <-> ( x = A /\ y = B ) )
7	6	anbi1i	|- ( ( <. A , B >. = <. x , y >. /\ ph ) <-> ( ( x = A /\ y = B ) /\ ph ) )
8	7	2exbii	|- ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> E. x E. y ( ( x = A /\ y = B ) /\ ph ) )
9		id	|- ( ( x = A /\ y = B ) -> ( x = A /\ y = B ) )
10	9 1	cgsex2g	|- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( ( x = A /\ y = B ) /\ ph ) <-> ps ) )
11	8 10	bitrid	|- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )