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Theorem indif 4286
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 4284 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
2 dfin4 4284 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
32difeq2i 4133 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 4278 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
51, 3, 43eqtr2i 2769 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cdif 3960  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980
This theorem is referenced by:  resdif  6870  kmlem11  10199  psgndiflemB  21636
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