Theorem ixpeq1d 6700

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 6699	. 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 1353   X_cixp 6688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-fn 5211  df-ixp 6689

This theorem is referenced by:  elixpsn  6725  ixpsnf1o  6726