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Theorem ssext 3948
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 3947 . . 3 (ABx(xAxB))
2 ssextss 3947 . . 3 (BAx(xBxA))
31, 2anbi12i 433 . 2 ((AB BA) ↔ (x(xAxB) x(xBxA)))
4 eqss 2954 . 2 (A = B ↔ (AB BA))
5 albiim 1373 . 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 1240   = wceq 1242  wss 2911
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373
This theorem is referenced by: (None)
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