Theorem oveqrspc2v 5924

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 2572	. 2 |- ( ph -> A. x e. A A. y e. B ( x F y ) = ( x G y ) )
3		oveq1 5904	. . . 4 |- ( x = X -> ( x F y ) = ( X F y ) )
4		oveq1 5904	. . . 4 |- ( x = X -> ( x G y ) = ( X G y ) )
5	3, 4	eqeq12d 2204	. . 3 |- ( x = X -> ( ( x F y ) = ( x G y ) <-> ( X F y ) = ( X G y ) ) )
6		oveq2 5905	. . . 4 |- ( y = Y -> ( X F y ) = ( X F Y ) )
7		oveq2 5905	. . . 4 |- ( y = Y -> ( X G y ) = ( X G Y ) )
8	6, 7	eqeq12d 2204	. . 3 |- ( y = Y -> ( ( X F y ) = ( X G y ) <-> ( X F Y ) = ( X G Y ) ) )
9	5, 8	rspc2v 2869	. 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 1364   e. wcel 2160   A.wral 2468  (class class class)co 5897

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900

This theorem is referenced by:  grpidpropdg  12853  sgrppropd  12891  mndpropd  12916  grpsubpropd2  13064  cmnpropd  13251  rngpropd  13326  ringpropd  13409  lmodprop2d  13681  lsspropdg  13764