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Theorem dfif3 4133
Description: Alternate definition of the conditional operator df-if 4120. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
Hypothesis
Ref Expression
dfif3.1 𝐶 = {𝑥𝜑}
Assertion
Ref Expression
dfif3 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem dfif3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfif6 4122 . 2 if(𝜑, 𝐴, 𝐵) = ({𝑦𝐴𝜑} ∪ {𝑦𝐵 ∣ ¬ 𝜑})
2 dfif3.1 . . . . . 6 𝐶 = {𝑥𝜑}
3 biidd 252 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜑))
43cbvabv 2776 . . . . . 6 {𝑥𝜑} = {𝑦𝜑}
52, 4eqtri 2673 . . . . 5 𝐶 = {𝑦𝜑}
65ineq2i 3844 . . . 4 (𝐴𝐶) = (𝐴 ∩ {𝑦𝜑})
7 dfrab3 3935 . . . 4 {𝑦𝐴𝜑} = (𝐴 ∩ {𝑦𝜑})
86, 7eqtr4i 2676 . . 3 (𝐴𝐶) = {𝑦𝐴𝜑}
9 dfrab3 3935 . . . 4 {𝑦𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑})
10 notab 3930 . . . . . 6 {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦𝜑})
115difeq2i 3758 . . . . . 6 (V ∖ 𝐶) = (V ∖ {𝑦𝜑})
1210, 11eqtr4i 2676 . . . . 5 {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶)
1312ineq2i 3844 . . . 4 (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶))
149, 13eqtr2i 2674 . . 3 (𝐵 ∩ (V ∖ 𝐶)) = {𝑦𝐵 ∣ ¬ 𝜑}
158, 14uneq12i 3798 . 2 ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦𝐴𝜑} ∪ {𝑦𝐵 ∣ ¬ 𝜑})
161, 15eqtr4i 2676 1 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  {cab 2637  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  cin 3606  ifcif 4119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-if 4120
This theorem is referenced by:  dfif4  4134
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