Theorem ixpeq1d 6597

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

Hypothesis

Ref	Expression
ixpeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
ixpeq1d	|- ( ph -> X_ x e. A C = X_ x e. B C )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   C( x)

Proof of Theorem ixpeq1d

Step	Hyp	Ref	Expression
1		ixpeq1d.1	. 2 |- ( ph -> A = B )
2		ixpeq1 6596	. 2 |- ( A = B -> X_ x e. A C = X_ x e. B C )
3	1, 2	syl 14	1 |- ( ph -> X_ x e. A C = X_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1331   X_cixp 6585

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-fn 5121  df-ixp 6586

This theorem is referenced by:  elixpsn  6622  ixpsnf1o  6623