Theorem mpteq2da 3846

Description: Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq2da.1	|- F/ x ph
mpteq2da.2	|- ( ( ph /\ x e. A ) -> B = C )

Assertion

Ref	Expression
mpteq2da	|- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )

Proof of Theorem mpteq2da

Step	Hyp	Ref	Expression
1		eqid 2040	. . 3 |- A = A
2	1	ax-gen 1338	. 2 |- A. x A = A
3		mpteq2da.1	. . 3 |- F/ x ph
4		mpteq2da.2	. . . 4 |- ( ( ph /\ x e. A ) -> B = C )
5	4	ex 108	. . 3 |- ( ph -> ( x e. A -> B = C ) )
6	3, 5	ralrimi 2390	. 2 |- ( ph -> A. x e. A B = C )
7		mpteq12f 3837	. 2 |- ( ( A. x A = A /\ A. x e. A B = C ) -> ( x e. A |-> B ) = ( x e. A |-> C ) )
8	2, 6, 7	sylancr 393	1 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   F/wnf 1349   e. wcel 1393   A.wral 2306   |-> cmpt 3818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-opab 3819  df-mpt 3820

This theorem is referenced by:  mpteq2dva  3847