Theorem mpt2eq3dva 5670

Description: Slightly more general equality inference for the maps to notation. (Contributed by set.mm contributors, 17-Oct-2013.) (Revised by set.mm contributors, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpt2eq3dva.1	|- ((ph /\ x e. A /\ y e. B) -> C = D)

Assertion

Ref	Expression
mpt2eq3dva	|- (ph -> (x e. A, y e. B |-> C) = (x e. A, y e. B |-> D))

Distinct variable groups:   ph,x   ph,y

Allowed substitution hints:   A(x,y)   B(x,y)   C(x,y)   D(x,y)

Proof of Theorem mpt2eq3dva

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpt2eq3dva.1	. . . . . 6 |- ((ph /\ x e. A /\ y e. B) -> C = D)
2	1	3expb 1152	. . . . 5 |- ((ph /\ (x e. A /\ y e. B)) -> C = D)
3	2	eqeq2d 2364	. . . 4 |- ((ph /\ (x e. A /\ y e. B)) -> (z = C <-> z = D))
4	3	pm5.32da 622	. . 3 |- (ph -> (((x e. A /\ y e. B) /\ z = C) <-> ((x e. A /\ y e. B) /\ z = D)))
5	4	oprabbidv 5565	. 2 |- (ph -> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = D)})
6		df-mpt2 5655	. 2 |- (x e. A, y e. B |-> C) = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
7		df-mpt2 5655	. 2 |- (x e. A, y e. B |-> D) = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = D)}
8	5, 6, 7	3eqtr4g 2410	1 |- (ph -> (x e. A, y e. B |-> C) = (x e. A, y e. B |-> D))

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  {coprab 5528   |-> cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-oprab 5529  df-mpt2 5655

This theorem is referenced by:  mpt2eq3ia  5671