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Metamath Proof Explorer

Theorem ixpeq2

Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006)

Ref Expression
Assertion ixpeq2 
|- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )

Proof

Step Hyp Ref Expression
1 ss2ixp 
 |-  ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )
2 ss2ixp 
 |-  ( A. x e. A C C_ B -> X_ x e. A C C_ X_ x e. A B )
3 1 2 anim12i 
 |-  ( ( A. x e. A B C_ C /\ A. x e. A C C_ B ) -> ( X_ x e. A B C_ X_ x e. A C /\ X_ x e. A C C_ X_ x e. A B ) )
4 eqss 
 |-  ( B = C <-> ( B C_ C /\ C C_ B ) )
5 4 ralbii 
 |-  ( A. x e. A B = C <-> A. x e. A ( B C_ C /\ C C_ B ) )
6 r19.26 
 |-  ( A. x e. A ( B C_ C /\ C C_ B ) <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
7 5 6 bitri 
 |-  ( A. x e. A B = C <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
8 eqss 
 |-  ( X_ x e. A B = X_ x e. A C <-> ( X_ x e. A B C_ X_ x e. A C /\ X_ x e. A C C_ X_ x e. A B ) )
9 3 7 8 3imtr4i 
 |-  ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )