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Theorem indif 4101
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 4099 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
2 dfin4 4099 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
32difeq2i 3954 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 4093 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
51, 3, 43eqtr2i 2855 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  cdif 3795  cin 3797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812
This theorem is referenced by:  resdif  6402  kmlem11  9304  psgndiflemB  20313
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