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Theorem dfif4 4322
 Description: Alternate definition of the conditional operator df-if 4308. 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 4321 . 2 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
3 undir 4103 . 2 ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))))
4 undi 4101 . . . 4 (𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶)))
5 undi 4101 . . . . 5 (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐶𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶)))
6 uncom 3980 . . . . . 6 (𝐶𝐵) = (𝐵𝐶)
7 unvdif 4266 . . . . . 6 (𝐶 ∪ (V ∖ 𝐶)) = V
86, 7ineq12i 4035 . . . . 5 ((𝐶𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) = ((𝐵𝐶) ∩ V)
9 inv1 4196 . . . . 5 ((𝐵𝐶) ∩ V) = (𝐵𝐶)
105, 8, 93eqtri 2806 . . . 4 (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = (𝐵𝐶)
114, 10ineq12i 4035 . . 3 ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = (((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵𝐶))
12 inass 4044 . . 3 (((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
1311, 12eqtri 2802 . 2 ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
142, 3, 133eqtri 2806 1 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601  {cab 2763  Vcvv 3398   ∖ cdif 3789   ∪ cun 3790   ∩ cin 3791  ifcif 4307 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308 This theorem is referenced by:  dfif5  4323
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