Theorem ssext 4206

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 4205	. . 3 |- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )
2		ssextss 4205	. . 3 |- ( B C_ A <-> A. x ( x C_ B -> x C_ A ) )
3	1, 2	anbi12i 457	. 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 3162	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		albiim 1480	. 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 211	1 |- ( A = B <-> A. x ( x C_ A <-> x C_ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by: (None)