MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfif3 Structured version   Visualization version   GIF version

Theorem dfif3 4453
Description: Alternate definition of the conditional operator df-if 4440. 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 4442 . 2 if(𝜑, 𝐴, 𝐵) = ({𝑦𝐴𝜑} ∪ {𝑦𝐵 ∣ ¬ 𝜑})
2 dfif3.1 . . . . . 6 𝐶 = {𝑥𝜑}
3 biidd 265 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜑))
43cbvabv 2890 . . . . . 6 {𝑥𝜑} = {𝑦𝜑}
52, 4eqtri 2845 . . . . 5 𝐶 = {𝑦𝜑}
65ineq2i 4160 . . . 4 (𝐴𝐶) = (𝐴 ∩ {𝑦𝜑})
7 dfrab3 4252 . . . 4 {𝑦𝐴𝜑} = (𝐴 ∩ {𝑦𝜑})
86, 7eqtr4i 2848 . . 3 (𝐴𝐶) = {𝑦𝐴𝜑}
9 dfrab3 4252 . . . 4 {𝑦𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑})
10 notab 4247 . . . . . 6 {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦𝜑})
115difeq2i 4071 . . . . . 6 (V ∖ 𝐶) = (V ∖ {𝑦𝜑})
1210, 11eqtr4i 2848 . . . . 5 {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶)
1312ineq2i 4160 . . . 4 (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶))
149, 13eqtr2i 2846 . . 3 (𝐵 ∩ (V ∖ 𝐶)) = {𝑦𝐵 ∣ ¬ 𝜑}
158, 14uneq12i 4112 . 2 ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦𝐴𝜑} ∪ {𝑦𝐵 ∣ ¬ 𝜑})
161, 15eqtr4i 2848 1 if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  {cab 2800  {crab 3134  Vcvv 3469  cdif 3905  cun 3906  cin 3907  ifcif 4439
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2178  ax-ext 2794
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 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-if 4440
This theorem is referenced by:  dfif4  4454
  Copyright terms: Public domain W3C validator