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Theorem ssext 3931
 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 = Bx(xAxB))
Distinct variable groups:   x,A   x,B

Proof of Theorem ssext
StepHypRef Expression
1 ssextss 3930 . . 3 (ABx(xAxB))
2 ssextss 3930 . . 3 (BAx(xBxA))
31, 2anbi12i 436 . 2 ((AB BA) ↔ (x(xAxB) x(xBxA)))
4 eqss 2937 . 2 (A = B ↔ (AB BA))
5 albiim 1357 . 2 (x(xAxB) ↔ (x(xAxB) x(xBxA)))
63, 4, 53bitr4i 201 1 (A = Bx(xAxB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1226   = wceq 1228   ⊆ wss 2894 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356 This theorem is referenced by: (None)
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