Theorem ixpeq2dva 6691

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

Hypothesis

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

Assertion

Ref	Expression
ixpeq2dva	|- ( ph -> X_ x e. A B = X_ x e. A C )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem ixpeq2dva

Step	Hyp	Ref	Expression
1		ixpeq2dva.1	. . 3 |- ( ( ph /\ x e. A ) -> B = C )
2	1	ralrimiva 2543	. 2 |- ( ph -> A. x e. A B = C )
3		ixpeq2 6690	. 2 |- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )
4	2, 3	syl 14	1 |- ( ph -> X_ x e. A B = X_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   X_cixp 6676

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-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-in 3127  df-ss 3134  df-ixp 6677

This theorem is referenced by:  ixpeq2dv  6692