Theorem oveqrspc2v 6028

Description: Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)

Hypothesis

Ref	Expression
oveqrspc2v.1	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x F y ) = ( x G y ) )

Assertion

Ref	Expression
oveqrspc2v	|- ( ( ph /\ ( X e. A /\ Y e. B ) ) -> ( X F Y ) = ( X G Y ) )

Distinct variable groups:   x, y, A   x, B, y   x, F, y   ph, x, y   y, Y   x, G, y   x, X, y

Allowed substitution hint:   Y( x)

Proof of Theorem oveqrspc2v

Step	Hyp	Ref	Expression
1		oveqrspc2v.1	. . 3 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x F y ) = ( x G y ) )
2	1	ralrimivva 2612	. 2 |- ( ph -> A. x e. A A. y e. B ( x F y ) = ( x G y ) )
3		oveq1 6008	. . . 4 |- ( x = X -> ( x F y ) = ( X F y ) )
4		oveq1 6008	. . . 4 |- ( x = X -> ( x G y ) = ( X G y ) )
5	3, 4	eqeq12d 2244	. . 3 |- ( x = X -> ( ( x F y ) = ( x G y ) <-> ( X F y ) = ( X G y ) ) )
6		oveq2 6009	. . . 4 |- ( y = Y -> ( X F y ) = ( X F Y ) )
7		oveq2 6009	. . . 4 |- ( y = Y -> ( X G y ) = ( X G Y ) )
8	6, 7	eqeq12d 2244	. . 3 |- ( y = Y -> ( ( X F y ) = ( X G y ) <-> ( X F Y ) = ( X G Y ) ) )
9	5, 8	rspc2v 2920	. 2 |- ( ( X e. A /\ Y e. B ) -> ( A. x e. A A. y e. B ( x F y ) = ( x G y ) -> ( X F Y ) = ( X G Y ) ) )
10	2, 9	mpan9 281	1 |- ( ( ph /\ ( X e. A /\ Y e. B ) ) -> ( X F Y ) = ( X G Y ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508  (class class class)co 6001

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  grpidpropdg  13407  sgrppropd  13446  mndpropd  13473  grpsubpropd2  13638  cmnpropd  13832  rngpropd  13918  ringpropd  14001  lmodprop2d  14312  lsspropdg  14395