Theorem ixpeq2dva 6607

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 2505	. 2 |- ( ph -> A. x e. A B = C )
3		ixpeq2 6606	. 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 1331   e. wcel 1480   A.wral 2416   X_cixp 6592

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-in 3077  df-ss 3084  df-ixp 6593

This theorem is referenced by:  ixpeq2dv  6608