Theorem ixpeq1d 6688

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 6687	. 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 1348   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-fn 5201  df-ixp 6677

This theorem is referenced by:  elixpsn  6713  ixpsnf1o  6714