Theorem mpteq12dva

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017) Remove dependency on ax-10 , ax-12 . (Revised by SN, 11-Nov-2024)

	Ref	Expression
Hypotheses	mpteq12dv.1	|- ( ph -> A = C )
	mpteq12dva.2	|- ( ( ph /\ x e. A ) -> B = D )
Assertion	mpteq12dva	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12dv.1	|- ( ph -> A = C )
2		mpteq12dva.2	|- ( ( ph /\ x e. A ) -> B = D )
3	2	eqeq2d	|- ( ( ph /\ x e. A ) -> ( y = B <-> y = D ) )
4	3	pm5.32da	|- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
5	1	eleq2d	|- ( ph -> ( x e. A <-> x e. C ) )
6	5	anbi1d	|- ( ph -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
7	4 6	bitrd	|- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = D ) ) )
8	7	opabbidv	|- ( ph -> { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = D ) } )
9		df-mpt	|- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
10		df-mpt	|- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
11	8 9 10	3eqtr4g	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )