Theorem ixpeq2 6799

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 6798	. . 3 |- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )
2		ss2ixp 6798	. . 3 |- ( A. x e. A C C_ B -> X_ x e. A C C_ X_ x e. A B )
3	1, 2	anim12i 338	. 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 3208	. . . 4 |- ( B = C <-> ( B C_ C /\ C C_ B ) )
5	4	ralbii 2512	. . 3 |- ( A. x e. A B = C <-> A. x e. A ( B C_ C /\ C C_ B ) )
6		r19.26 2632	. . 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 184	. 2 |- ( A. x e. A B = C <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
8		eqss 3208	. 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 201	1 |- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   A.wral 2484   C_ wss 3166   X_cixp 6785

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-in 3172  df-ss 3179  df-ixp 6786

This theorem is referenced by:  ixpeq2dva  6800  ixpintm  6812  prdsbas3  13119  pwsbas  13124