Theorem mpt2eq3ia 5706

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpt2eq3ia.1	|- ( ( x e. A /\ y e. B ) -> C = D )

Assertion

Ref	Expression
mpt2eq3ia	|- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )

Proof of Theorem mpt2eq3ia

Step	Hyp	Ref	Expression
1		mpt2eq3ia.1	. . . 4 |- ( ( x e. A /\ y e. B ) -> C = D )
2	1	3adant1 961	. . 3 |- ( ( T. /\ x e. A /\ y e. B ) -> C = D )
3	2	mpt2eq3dva 5705	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )
4	3	mptru 1298	1 |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   T. wtru 1290   e. wcel 1438   |-> cmpt2 5646

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-oprab 5648  df-mpt2 5649

This theorem is referenced by:  oprab2co  5975  genpdf  7057  dfioo2  9382  iseqvalcbv  9860