Metamath Proof Explorer

Theorem ixpeq2dva

Description: Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016)

Ref Expression
Hypothesis ixpeq2dva.1 
|- ( ( ph /\ x e. A ) -> B = C )
Assertion ixpeq2dva 
|- ( ph -> X_ x e. A B = X_ x e. A C )

Proof

Step Hyp Ref Expression
1 ixpeq2dva.1 
 |-  ( ( ph /\ x e. A ) -> B = C )
2 1 ralrimiva 
 |-  ( ph -> A. x e. A B = C )
3 ixpeq2 
 |-  ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )
4 2 3 syl 
 |-  ( ph -> X_ x e. A B = X_ x e. A C )