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Theorem dfif4 4245
Description: Alternate definition of the conditional operator df-if 4231. 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 4244 . 2 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
3 undir 4019 . 2 ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))))
4 undi 4017 . . . 4 (𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶)))
5 undi 4017 . . . . 5 (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐶𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶)))
6 uncom 3900 . . . . . 6 (𝐶𝐵) = (𝐵𝐶)
7 unvdif 4186 . . . . . 6 (𝐶 ∪ (V ∖ 𝐶)) = V
86, 7ineq12i 3955 . . . . 5 ((𝐶𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) = ((𝐵𝐶) ∩ V)
9 inv1 4113 . . . . 5 ((𝐵𝐶) ∩ V) = (𝐵𝐶)
105, 8, 93eqtri 2786 . . . 4 (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = (𝐵𝐶)
114, 10ineq12i 3955 . . 3 ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = (((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵𝐶))
12 inass 3966 . . 3 (((𝐴𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
1311, 12eqtri 2782 . 2 ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
142, 3, 133eqtri 2786 1 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  {cab 2746  Vcvv 3340  cdif 3712  cun 3713  cin 3714  ifcif 4230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231
This theorem is referenced by:  dfif5  4246
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