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Metamath Proof Explorer

Theorem caovclg

Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014)

Ref Expression
Hypothesis caovclg.1 
|- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E )
Assertion caovclg 
|- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )

Proof

Step Hyp Ref Expression
1 caovclg.1 
 |-  ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E )
2 1 ralrimivva 
 |-  ( ph -> A. x e. C A. y e. D ( x F y ) e. E )
3 oveq1 
 |-  ( x = A -> ( x F y ) = ( A F y ) )
4 3 eleq1d 
 |-  ( x = A -> ( ( x F y ) e. E <-> ( A F y ) e. E ) )
5 oveq2 
 |-  ( y = B -> ( A F y ) = ( A F B ) )
6 5 eleq1d 
 |-  ( y = B -> ( ( A F y ) e. E <-> ( A F B ) e. E ) )
7 4 6 rspc2v 
 |-  ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ( x F y ) e. E -> ( A F B ) e. E ) )
8 2 7 mpan9 
 |-  ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )