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Theorem ssofeq 4077
 Description: When A and B are subsets of C, equality depends only on the elements of C. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
ssofeq ((A C B C) → (A = Bx C (x Ax B)))
Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem ssofeq
StepHypRef Expression
1 ssofss 4076 . . 3 (A C → (A Bx C (x Ax B)))
2 ssofss 4076 . . 3 (B C → (B Ax C (x Bx A)))
31, 2bi2anan9 843 . 2 ((A C B C) → ((A B B A) ↔ (x C (x Ax B) x C (x Bx A))))
4 eqss 3287 . 2 (A = B ↔ (A B B A))
5 ralbiim 2751 . 2 (x C (x Ax B) ↔ (x C (x Ax B) x C (x Bx A)))
63, 4, 53bitr4g 279 1 ((A C B C) → (A = Bx C (x Ax B)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259 This theorem is referenced by:  eqpw1  4162  pw111  4170  eqrelk  4212  sikexlem  4295  insklem  4304
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