Theorem ixpeq2d 39237

Description: Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
ixpeq2d.1	|- F/ x ph
ixpeq2d.2	|- ( ( ph /\ x e. A ) -> B = C )

Assertion

Ref	Expression
ixpeq2d	|- ( ph -> X_ x e. A B = X_ x e. A C )

Proof of Theorem ixpeq2d

Step	Hyp	Ref	Expression
1		ixpeq2d.1	. . 3 |- F/ x ph
2		ixpeq2d.2	. . . 4 |- ( ( ph /\ x e. A ) -> B = C )
3	2	ex 450	. . 3 |- ( ph -> ( x e. A -> B = C ) )
4	1, 3	ralrimi 2957	. 2 |- ( ph -> A. x e. A B = C )
5		ixpeq2 7922	. 2 |- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )
6	4, 5	syl 17	1 |- ( ph -> X_ x e. A B = X_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-in 3581  df-ss 3588  df-ixp 7909

This theorem is referenced by:  hoicvrrex  40770  ovnlecvr  40772  ovnhoilem1  40815  hoi2toco  40821  ovnlecvr2  40824  opnvonmbllem1  40846