Theorem mpt2eq3ia 5512

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 921	. . 3 |- (( T. /\ x e. A /\ y e. B) -> C = D)
3	2	mpt2eq3dva 5511	. 2 |- ( T. -> (x e. A , y e. B |-> C) = (x e. A , y e. B |-> D))
4	3	trud 1251	1 |- (x e. A , y e. B |-> C) = (x e. A , y e. B |-> D)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   T. wtru 1243   e. wcel 1390   |-> cmpt2 5457

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-oprab 5459  df-mpt2 5460

This theorem is referenced by:  oprab2co  5781  genpdf  6490  dfioo2  8573