Theorem oveqrspc2v 5994

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 2590	. 2 |- ( ph -> A. x e. A A. y e. B ( x F y ) = ( x G y ) )
3		oveq1 5974	. . . 4 |- ( x = X -> ( x F y ) = ( X F y ) )
4		oveq1 5974	. . . 4 |- ( x = X -> ( x G y ) = ( X G y ) )
5	3, 4	eqeq12d 2222	. . 3 |- ( x = X -> ( ( x F y ) = ( x G y ) <-> ( X F y ) = ( X G y ) ) )
6		oveq2 5975	. . . 4 |- ( y = Y -> ( X F y ) = ( X F Y ) )
7		oveq2 5975	. . . 4 |- ( y = Y -> ( X G y ) = ( X G Y ) )
8	6, 7	eqeq12d 2222	. . 3 |- ( y = Y -> ( ( X F y ) = ( X G y ) <-> ( X F Y ) = ( X G Y ) ) )
9	5, 8	rspc2v 2897	. 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 1373   e. wcel 2178   A.wral 2486  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  grpidpropdg  13321  sgrppropd  13360  mndpropd  13387  grpsubpropd2  13552  cmnpropd  13746  rngpropd  13832  ringpropd  13915  lmodprop2d  14225  lsspropdg  14308