Theorem mpteq2dv 3904

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)

Hypothesis

Ref	Expression
mpteq2dv.1	|- ( ph -> B = C )

Assertion

Ref	Expression
mpteq2dv	|- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )

Distinct variable group:   ph, x

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

Proof of Theorem mpteq2dv

Step	Hyp	Ref	Expression
1		mpteq2dv.1	. . 3 |- ( ph -> B = C )
2	1	adantr 270	. 2 |- ( ( ph /\ x e. A ) -> B = C )
3	2	mpteq2dva 3903	1 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )

Syntax hints:   -> wi 4   = wceq 1287   e. wcel 1436   |-> cmpt 3874

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-ral 2360  df-opab 3875  df-mpt 3876

This theorem is referenced by:  ofeq  5809  rdgeq1  6084  rdgeq2  6085  omv  6164  oeiv  6165  0tonninf  9766  1tonninf  9767  iseqf1olemjpcl  9821  iseqf1olemqpcl  9822  iseqf1olemfvp  9823  iseqf1olemqsum  9826  iseqf1olemp  9828  isummo  10655  zisum  10656  fisum  10658  peano4nninf  11326  peano3nninf  11327  nninfalllem1  11329  nninfsellemdc  11332  nninfsellemeq  11336  nninfsellemqall  11337  nninfsellemeqinf  11338  nninfomni  11341