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Theorem difeqri 4085
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 3921 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 difeqri.1 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 275 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2730 1 (𝐴𝐵) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397   = wceq 1542  wcel 2107  cdif 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-dif 3914
This theorem is referenced by:  difdif  4091  ddif  4097  dfss4  4219  difin  4222  difab  4261
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