Theorem mpteq12dv 4069

Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq12dv.1	|- ( ph -> A = C )
mpteq12dv.2	|- ( ph -> B = D )

Assertion

Ref	Expression
mpteq12dv	|- ( 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 mpteq12dv

Step	Hyp	Ref	Expression
1		mpteq12dv.1	. 2 |- ( ph -> A = C )
2		mpteq12dv.2	. . 3 |- ( ph -> B = D )
3	2	adantr 274	. 2 |- ( ( ph /\ x e. A ) -> B = D )
4	1, 3	mpteq12dva 4068	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   |-> cmpt 4048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-opab 4049  df-mpt 4050

This theorem is referenced by:  mpteq12i  4075  offval  6066  offval3  6111  odzval  12188  restval  12578  ntrfval  12859  clsfval  12860  neifval  12899  cnpfval  12954  cnprcl2k  12965  reldvg  13407  dvfvalap  13409  eldvap  13410