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Theorem dfif4 4465
Description: Alternate definition of the conditional operator df-if 4451. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
Hypothesis
Ref Expression
dfif3.1 𝐶 = {𝑥𝜑}
Assertion
Ref Expression
dfif4 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem dfif4
StepHypRef Expression
1 dfif3.1 . . 3 𝐶 = {𝑥𝜑}
21dfif3 4464 . 2 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
3 undir 4238 . 2 ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))))
4 undi 4236 . . . 4 (𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶)))
5 undi 4236 . . . . 5 (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐶𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶)))
6 uncom 4115 . . . . . 6 (𝐶𝐵) = (𝐵𝐶)
7 unvdif 4406 . . . . . 6 (𝐶 ∪ (V ∖ 𝐶)) = V
86, 7ineq12i 4172 . . . . 5 ((𝐶𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) = ((𝐵𝐶) ∩ V)
9 inv1 4331 . . . . 5 ((𝐵𝐶) ∩ V) = (𝐵𝐶)
105, 8, 93eqtri 2851 . . . 4 (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = (𝐵𝐶)
114, 10ineq12i 4172 . . 3 ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = (((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵𝐶))
12 inass 4181 . . 3 (((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
1311, 12eqtri 2847 . 2 ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
142, 3, 133eqtri 2851 1 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  {cab 2802  Vcvv 3480  cdif 3916  cun 3917  cin 3918  ifcif 4450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451
This theorem is referenced by:  dfif5  4466
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