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Theorem indif 4203
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 4201 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
2 dfin4 4201 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
32difeq2i 4053 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 4195 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
51, 3, 43eqtr2i 2772 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cdif 3883  cin 3885
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3431  df-dif 3889  df-in 3893  df-ss 3903
This theorem is referenced by:  resdif  6729  kmlem11  9926  psgndiflemB  20815
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