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Theorem difdif 3046
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif (A ∖ (BA)) = A

Proof of Theorem difdif
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ax-ia1 99 . . 3 ((x A ¬ x (BA)) → x A)
2 pm4.45im 317 . . . 4 (x A ↔ (x A (x Bx A)))
3 imanim 778 . . . . . 6 ((x Bx A) → ¬ (x B ¬ x A))
4 eldif 2904 . . . . . 6 (x (BA) ↔ (x B ¬ x A))
53, 4sylnibr 589 . . . . 5 ((x Bx A) → ¬ x (BA))
65anim2i 324 . . . 4 ((x A (x Bx A)) → (x A ¬ x (BA)))
72, 6sylbi 114 . . 3 (x A → (x A ¬ x (BA)))
81, 7impbii 117 . 2 ((x A ¬ x (BA)) ↔ x A)
98difeqri 3041 1 (A ∖ (BA)) = A
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = 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:  dif0  3271
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