Theorem mpteq12dva 4099

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 1885	. 2 |- ( ph -> A. x A = C )
3		mpteq12dva.2	. . 3 |- ( ( ph /\ x e. A ) -> B = D )
4	3	ralrimiva 2563	. 2 |- ( ph -> A. x e. A B = D )
5		mpteq12f 4098	. 2 |- ( ( A. x A = C /\ A. x e. A B = D ) -> ( x e. A |-> B ) = ( x e. C |-> D ) )
6	2, 4, 5	syl2anc 411	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   e. wcel 2160   A.wral 2468   |-> cmpt 4079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-opab 4080  df-mpt 4081

This theorem is referenced by:  mpteq12dv  4100