Theorem ixpeq2 6678

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 6677	. . 3 |- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )
2		ss2ixp 6677	. . 3 |- ( A. x e. A C C_ B -> X_ x e. A C C_ X_ x e. A B )
3	1, 2	anim12i 336	. 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 3157	. . . 4 |- ( B = C <-> ( B C_ C /\ C C_ B ) )
5	4	ralbii 2472	. . 3 |- ( A. x e. A B = C <-> A. x e. A ( B C_ C /\ C C_ B ) )
6		r19.26 2592	. . 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 183	. 2 |- ( A. x e. A B = C <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
8		eqss 3157	. 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 200	1 |- ( A. x e. A B = C -> X_ x e. A B = X_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   A.wral 2444   C_ wss 3116   X_cixp 6664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-in 3122  df-ss 3129  df-ixp 6665

This theorem is referenced by:  ixpeq2dva  6679  ixpintm  6691