Theorem ixpeq1d 6922

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 6921	. 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 1398   X_cixp 6910

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-fn 5336  df-ixp 6911

This theorem is referenced by:  elixpsn  6947  ixpsnf1o  6948  ptex  13427  prdsex  13432  prdsval  13436