Theorem ixpeq2dva 6925

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

Hypothesis

Ref	Expression
ixpeq2dva.1	|- ( ( ph /\ x e. A ) -> B = C )

Assertion

Ref	Expression
ixpeq2dva	|- ( ph -> X_ x e. A B = X_ x e. A C )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem ixpeq2dva

Step	Hyp	Ref	Expression
1		ixpeq2dva.1	. . 3 |- ( ( ph /\ x e. A ) -> B = C )
2	1	ralrimiva 2606	. 2 |- ( ph -> A. x e. A B = C )
3		ixpeq2 6924	. 2 |- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )
4	2, 3	syl 14	1 |- ( ph -> X_ x e. A B = X_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   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-in 3207  df-ss 3214  df-ixp 6911

This theorem is referenced by:  ixpeq2dv  6926