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Theorem indif 3902
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
StepHypRef Expression
1 dfin4 3900 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
2 dfin4 3900 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
32difeq2i 3758 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 3894 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
51, 3, 43eqtr2i 2679 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cdif 3604  cin 3606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621
This theorem is referenced by:  resdif  6195  kmlem11  9020  psgndiflemB  19994
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