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Theorem dfif3 3455
 Description: Alternate definition of the conditional operator df-if 3443. 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 3444 . 2 if(𝜑, 𝐴, 𝐵) = ({𝑦𝐴𝜑} ∪ {𝑦𝐵 ∣ ¬ 𝜑})
2 dfif3.1 . . . . . 6 𝐶 = {𝑥𝜑}
3 biidd 171 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜑))
43cbvabv 2239 . . . . . 6 {𝑥𝜑} = {𝑦𝜑}
52, 4eqtri 2136 . . . . 5 𝐶 = {𝑦𝜑}
65ineq2i 3242 . . . 4 (𝐴𝐶) = (𝐴 ∩ {𝑦𝜑})
7 dfrab3 3320 . . . 4 {𝑦𝐴𝜑} = (𝐴 ∩ {𝑦𝜑})
86, 7eqtr4i 2139 . . 3 (𝐴𝐶) = {𝑦𝐴𝜑}
9 dfrab3 3320 . . . 4 {𝑦𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑})
10 notab 3314 . . . . . 6 {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦𝜑})
115difeq2i 3159 . . . . . 6 (V ∖ 𝐶) = (V ∖ {𝑦𝜑})
1210, 11eqtr4i 2139 . . . . 5 {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶)
1312ineq2i 3242 . . . 4 (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶))
149, 13eqtr2i 2137 . . 3 (𝐵 ∩ (V ∖ 𝐶)) = {𝑦𝐵 ∣ ¬ 𝜑}
158, 14uneq12i 3196 . 2 ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦𝐴𝜑} ∪ {𝑦𝐵 ∣ ¬ 𝜑})
161, 15eqtr4i 2139 1 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1314  {cab 2101  {crab 2395  Vcvv 2658   ∖ cdif 3036   ∪ cun 3037   ∩ cin 3038  ifcif 3442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-if 3443 This theorem is referenced by: (None)
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