Theorem mpteq12dva 4480

Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
mpteq12dva	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)   C( x)   D( x)

Proof of Theorem mpteq12dva

Step	Hyp	Ref	Expression
1		mpteq12dv.1	. . 3 |- ( ph -> A = C )
2	1	alrimiv 1686	. 2 |- ( ph -> A. x A = C )
3		mpteq12dva.2	. . 3 |- ( ( ph /\ x e. A ) -> B = D )
4	3	ralrimiva 2830	. 2 |- ( ph -> A. x e. A B = D )
5		mpteq12f 4479	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
6	2, 4, 5	syl2anc 661	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1368   = wceq 1370   e. wcel 1758   A.wral 2799   |-> cmpt 4461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-ral 2804  df-opab 4462  df-mpt 4463

This theorem is referenced by:  mpteq12dv  4481  reps  12530  repswccat  12545  cidpropd  14772  monpropd  14799  fucpropd  15010  curfpropd  15166  hofpropd  15200  yonffthlem  15215  ofco2  18469  rrxnm  21037  sgnsv  26362  ofcfval  26708  ccatmulgnn0dir  27107  signstf0  27136  pmatcollpw3fi1lem1  31296