Theorem mpt2eq3dva 5589

Description: Slightly more general equality inference for the maps to notation. (Contributed by NM, 17-Oct-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 1139	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = D )
3	2	eqeq2d 2092	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( z = C <-> z = D ) )
4	3	pm5.32da 439	. . 3 |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ z = D ) ) )
5	4	oprabbidv 5579	. 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 5537	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7		df-mpt2 5537	. 2 |- ( x e. A , y e. B |-> D ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = D ) }
8	5, 6, 7	3eqtr4g 2138	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   {coprab 5533   |-> cmpt2 5534

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  mpt2eq3ia  5590  ofeq  5734  fmpt2co  5857  iseqeq2  9435  iseqeq3  9436  iseqval  9440