Theorem mpteq12dv 4197

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 276	. 2 |- ( ( ph /\ x e. A ) -> B = D )
4	1, 3	mpteq12dva 4196	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   |-> cmpt 4176

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-ral 2527  df-opab 4177  df-mpt 4178

This theorem is referenced by:  mpteq12i  4203  offval  6283  offval3  6340  ccatfvalfi  11305  swrdval  11365  odzval  12964  restval  13542  qusval  13587  grpinvfvalg  13797  grpinvpropdg  13830  prdsex  14114  prdsval  14115  opprnegg  14327  lspfval  14662  lsppropd  14706  sraval  14711  psrval  14940  ntrfval  15091  clsfval  15092  neifval  15131  cnpfval  15186  cnprcl2k  15197  reldvg  15670  dvfvalap  15672  eldvap  15673  vtxdgfval  16409  vtxdgop  16413  vtxdeqd  16417