Theorem oveqrspc2v 5880

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 2552	. 2 |- ( ph -> A. x e. A A. y e. B ( x F y ) = ( x G y ) )
3		oveq1 5860	. . . 4 |- ( x = X -> ( x F y ) = ( X F y ) )
4		oveq1 5860	. . . 4 |- ( x = X -> ( x G y ) = ( X G y ) )
5	3, 4	eqeq12d 2185	. . 3 |- ( x = X -> ( ( x F y ) = ( x G y ) <-> ( X F y ) = ( X G y ) ) )
6		oveq2 5861	. . . 4 |- ( y = Y -> ( X F y ) = ( X F Y ) )
7		oveq2 5861	. . . 4 |- ( y = Y -> ( X G y ) = ( X G Y ) )
8	6, 7	eqeq12d 2185	. . 3 |- ( y = Y -> ( ( X F y ) = ( X G y ) <-> ( X F Y ) = ( X G Y ) ) )
9	5, 8	rspc2v 2847	. 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 279	1 |- ( ( ph /\ ( X e. A /\ Y e. B ) ) -> ( X F Y ) = ( X G Y ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448  (class class class)co 5853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  grpidpropdg  12628  mndpropd  12676