Theorem mpteq12dv 4213

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 4212	1 |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   |-> cmpt 4192

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-opab 4193  df-mpt 4194

This theorem is used by:  mpteq12i  4219  offval  6310  offval3  6367  indv  9295  ccatfvalfi  11360  swrdval  11420  odzval  13020  restval  13599  qusval  13644  grpinvfvalg  13847  grpinvpropdg  13880  prdsex  14172  prdsval  14173  opprnegg  14389  lspfval  14725  lsppropd  14769  sraval  14774  aspval  15015  asclfval  15021  ressascl  15039  psrval  15050  ntrfval  15201  clsfval  15202  neifval  15241  cnpfval  15296  cnprcl2k  15307  reldvg  15780  dvfvalap  15782  eldvap  15783  vtxdgfval  16529  vtxdgop  16533  vtxdeqd  16537