Theorem ixpeq2 6812

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 6811	. . 3 |- ( A. x e. A B C_ C -> X_ x e. A B C_ X_ x e. A C )
2		ss2ixp 6811	. . 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 3212	. . . 4 |- ( B = C <-> ( B C_ C /\ C C_ B ) )
5	4	ralbii 2513	. . 3 |- ( A. x e. A B = C <-> A. x e. A ( B C_ C /\ C C_ B ) )
6		r19.26 2633	. . 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 3212	. 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 2485   C_ wss 3170   X_cixp 6798

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-in 3176  df-ss 3183  df-ixp 6799

This theorem is referenced by:  ixpeq2dva  6813  ixpintm  6825  prdsbas3  13194  pwsbas  13199