Theorem ssext 4223

Description: An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)

Assertion

Ref	Expression
ssext	|- ( A = B <-> A. x ( x C_ A <-> x C_ B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem ssext

Step	Hyp	Ref	Expression
1		ssextss 4222	. . 3 |- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )
2		ssextss 4222	. . 3 |- ( B C_ A <-> A. x ( x C_ B -> x C_ A ) )
3	1, 2	anbi12i 460	. 2 |- ( ( A C_ B /\ B C_ A ) <-> ( A. x ( x C_ A -> x C_ B ) /\ A. x ( x C_ B -> x C_ A ) ) )
4		eqss 3172	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		albiim 1487	. 2 |- ( A. x ( x C_ A <-> x C_ B ) <-> ( A. x ( x C_ A -> x C_ B ) /\ A. x ( x C_ B -> x C_ A ) ) )
6	3, 4, 5	3bitr4i 212	1 |- ( A = B <-> A. x ( x C_ A <-> x C_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   C_ wss 3131

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600

This theorem is referenced by: (None)