Theorem oveqrspc2v 5952

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 2579	. 2 |- ( ph -> A. x e. A A. y e. B ( x F y ) = ( x G y ) )
3		oveq1 5932	. . . 4 |- ( x = X -> ( x F y ) = ( X F y ) )
4		oveq1 5932	. . . 4 |- ( x = X -> ( x G y ) = ( X G y ) )
5	3, 4	eqeq12d 2211	. . 3 |- ( x = X -> ( ( x F y ) = ( x G y ) <-> ( X F y ) = ( X G y ) ) )
6		oveq2 5933	. . . 4 |- ( y = Y -> ( X F y ) = ( X F Y ) )
7		oveq2 5933	. . . 4 |- ( y = Y -> ( X G y ) = ( X G Y ) )
8	6, 7	eqeq12d 2211	. . 3 |- ( y = Y -> ( ( X F y ) = ( X G y ) <-> ( X F Y ) = ( X G Y ) ) )
9	5, 8	rspc2v 2881	. 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 2167   A.wral 2475  (class class class)co 5925

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  grpidpropdg  13076  sgrppropd  13115  mndpropd  13142  grpsubpropd2  13307  cmnpropd  13501  rngpropd  13587  ringpropd  13670  lmodprop2d  13980  lsspropdg  14063