Theorem ixpeq1d 6992

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 6991	. 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 )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   X_cixp 6980

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-fn 5380  df-ixp 6981

This theorem is used by:  elixpsn  7017  ixpsnf1o  7018  ptex  13618  prdsex  14172  prdsval  14173