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Theorem indif 3180
 Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
indif (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem indif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 anabs5 507 . . 3 ((𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 elin 3126 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝐴𝐵)))
3 eldif 2927 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
43anbi2i 430 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
52, 4bitri 173 . . 3 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ (𝑥𝐴 ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)))
61, 5, 33bitr4i 201 . 2 (𝑥 ∈ (𝐴 ∩ (𝐴𝐵)) ↔ 𝑥 ∈ (𝐴𝐵))
76eqriv 2037 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ∖ cdif 2914   ∩ cin 2916 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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924 This theorem is referenced by:  resdif  5148
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