Metamath Proof Explorer

Theorem xpeq12

Description: Equality theorem for Cartesian product. (Contributed by FL, 31-Aug-2009)

Ref Expression
Assertion xpeq12 
|- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )

Proof

Step Hyp Ref Expression
1 xpeq1 
 |-  ( A = B -> ( A X. C ) = ( B X. C ) )
2 xpeq2 
 |-  ( C = D -> ( B X. C ) = ( B X. D ) )
3 1 2 sylan9eq 
 |-  ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )