Theorem mpteq2da 4178

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 2231	. . 3 |- A = A
2	1	ax-gen 1497	. 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 115	. . 3 |- ( ph -> ( x e. A -> B = C ) )
6	3, 5	ralrimi 2603	. 2 |- ( ph -> A. x e. A B = C )
7		mpteq12f 4169	. 2 |- ( ( A. x A = A /\ A. x e. A B = C ) -> ( x e. A |-> B ) = ( x e. A |-> C ) )
8	2, 6, 7	sylancr 414	1 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1395   = wceq 1397   F/wnf 1508   e. wcel 2202   A.wral 2510   |-> cmpt 4150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-opab 4151  df-mpt 4152

This theorem is referenced by:  mpteq2dva  4179  prodeq1f  12112  prodeq2  12117  gsumfzsnfd  13931