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Theorem difeqri 3387
 Description: Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypothesis
Ref Expression
difeqri.1 ((x A ¬ x B) ↔ x C)
Assertion
Ref Expression
difeqri (A B) = C
Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem difeqri
StepHypRef Expression
1 eldif 3221 . . 3 (x (A B) ↔ (x A ¬ x B))
2 difeqri.1 . . 3 ((x A ¬ x B) ↔ x C)
31, 2bitri 240 . 2 (x (A B) ↔ x C)
43eqriv 2350 1 (A B) = C
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215 This theorem is referenced by:  difdif  3392  ddif  3398  dfss4  3489  difin  3492  difab  3523
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