Theorem mpteq2da 4071

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 2165	. . 3 |- A = A
2	1	ax-gen 1437	. 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 114	. . 3 |- ( ph -> ( x e. A -> B = C ) )
6	3, 5	ralrimi 2537	. 2 |- ( ph -> A. x e. A B = C )
7		mpteq12f 4062	. 2 |- ( ( A. x A = A /\ A. x e. A B = C ) -> ( x e. A |-> B ) = ( x e. A |-> C ) )
8	2, 6, 7	sylancr 411	1 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   = wceq 1343   F/wnf 1448   e. wcel 2136   A.wral 2444   |-> cmpt 4043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-opab 4044  df-mpt 4045

This theorem is referenced by:  mpteq2dva  4072  prodeq1f  11493  prodeq2  11498