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Theorem difeqri 3722
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 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)
Assertion
Ref Expression
difeqri (𝐴𝐵) = 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem difeqri
StepHypRef Expression
1 eldif 3577 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 difeqri.1 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 264 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2617 1 (𝐴𝐵) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384   = wceq 1481  wcel 1988  cdif 3564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570
This theorem is referenced by:  difdif  3728  ddif  3734  dfss4  3850  difin  3853  difab  3888
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