Theorem mpteq2dv 3895

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 3894	1 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   |-> cmpt 3865

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-opab 3866  df-mpt 3867

This theorem is referenced by:  ofeq  5793  rdgeq1  6068  rdgeq2  6069  omv  6148  oeiv  6149  0tonninf  9734  1tonninf  9735