MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfif2 Structured version   Visualization version   GIF version

Theorem dfif2 4427
Description: An alternate definition of the conditional operator df-if 4426 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
Assertion
Ref Expression
dfif2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfif2
StepHypRef Expression
1 df-if 4426 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 df-or 845 . . . 4 (((𝑥𝐵 ∧ ¬ 𝜑) ∨ (𝑥𝐴𝜑)) ↔ (¬ (𝑥𝐵 ∧ ¬ 𝜑) → (𝑥𝐴𝜑)))
3 orcom 867 . . . 4 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐵 ∧ ¬ 𝜑) ∨ (𝑥𝐴𝜑)))
4 iman 405 . . . . 5 ((𝑥𝐵𝜑) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝜑))
54imbi1i 353 . . . 4 (((𝑥𝐵𝜑) → (𝑥𝐴𝜑)) ↔ (¬ (𝑥𝐵 ∧ ¬ 𝜑) → (𝑥𝐴𝜑)))
62, 3, 53bitr4i 306 . . 3 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
76abbii 2863 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
81, 7eqtri 2821 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2111  {cab 2776  ifcif 4425
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-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-if 4426
This theorem is referenced by:  iftrue  4431  nfifd  4453
  Copyright terms: Public domain W3C validator