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Theorem difeqri 3041
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 (AB) = 𝐶
Distinct variable groups:   x,A   x,B   x,𝐶

Proof of Theorem difeqri
StepHypRef Expression
1 eldif 2904 . . 3 (x (AB) ↔ (x A ¬ x B))
2 difeqri.1 . . 3 ((x A ¬ x B) ↔ x 𝐶)
31, 2bitri 173 . 2 (x (AB) ↔ x 𝐶)
43eqriv 2019 1 (AB) = 𝐶
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98   = wceq 1228   wcel 1374  cdif 2891
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-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897
This theorem is referenced by:  difdif  3046  ddifnel  3052  difab  3183
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