Theorem mpteq2da 4144

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 2206	. . 3 |- A = A
2	1	ax-gen 1473	. 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 2578	. 2 |- ( ph -> A. x e. A B = C )
7		mpteq12f 4135	. 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 1371   = wceq 1373   F/wnf 1484   e. wcel 2177   A.wral 2485   |-> cmpt 4116

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-ral 2490  df-opab 4117  df-mpt 4118

This theorem is referenced by:  mpteq2dva  4145  prodeq1f  11948  prodeq2  11953  gsumfzsnfd  13766