Theorem ixpeq2 7922

Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Assertion

Ref	Expression
ixpeq2	|- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )

Proof of Theorem ixpeq2

Step	Hyp	Ref	Expression
1		ss2ixp 7921	. . 3 |- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )
2		ss2ixp 7921	. . 3 |- ( A. x e. A C C_ B -> X_ x e. A C C_ X_ x e. A B )
3	1, 2	anim12i 590	. 2 |- ( ( A. x e. A B C_ C /\ A. x e. A C C_ B ) -> ( X_ x e. A B C_ X_ x e. A C /\ X_ x e. A C C_ X_ x e. A B ) )
4		eqss 3618	. . . 4 |- ( B = C <-> ( B C_ C /\ C C_ B ) )
5	4	ralbii 2980	. . 3 |- ( A. x e. A B = C <-> A. x e. A ( B C_ C /\ C C_ B ) )
6		r19.26 3064	. . 3 |- ( A. x e. A ( B C_ C /\ C C_ B ) <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
7	5, 6	bitri 264	. 2 |- ( A. x e. A B = C <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
8		eqss 3618	. 2 |- ( X_ x e. A B = X_ x e. A C <-> ( X_ x e. A B C_ X_ x e. A C /\ X_ x e. A C C_ X_ x e. A B ) )
9	3, 7, 8	3imtr4i 281	1 |- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   C_ wss 3574   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:  ixpeq2dva  7923  ixpint  7935  prdsbas3  16141  pwsbas  16147  ptbasfi  21384  ptunimpt  21398  pttopon  21399  ptcld  21416  ptrescn  21442  ptuncnv  21610  ptunhmeo  21611  ptrest  33408  prdstotbnd  33593  ixpeq2d  39237  hoidmv1le  40808