Theorem copsex2gd

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) Adapt copsex2g $p to deduction form. (Revised by BJ, 28-Mar-2026) Do not use copsex2g . (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	copsex2gd.is	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
Assertion	copsex2gd	|- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		copsex2gd.is	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
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 >. /\ ps ) <-> ( ( x = A /\ y = B ) /\ ps ) )
8	7	2exbii	|- ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> E. x E. y ( ( x = A /\ y = B ) /\ ps ) )
9		simpr	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( x = A /\ y = B ) )
10	9 1	cgsex2gd	|- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( E. x E. y ( ( x = A /\ y = B ) /\ ps ) <-> ch ) )
11	8 10	bitrid	|- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ch ) )