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Theorem dfif2 4062
Description: An alternate definition of the conditional operator df-if 4061 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 4061 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 df-or 385 . . . 4 (((𝑥𝐵 ∧ ¬ 𝜑) ∨ (𝑥𝐴𝜑)) ↔ (¬ (𝑥𝐵 ∧ ¬ 𝜑) → (𝑥𝐴𝜑)))
3 orcom 402 . . . 4 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐵 ∧ ¬ 𝜑) ∨ (𝑥𝐴𝜑)))
4 iman 440 . . . . 5 ((𝑥𝐵𝜑) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝜑))
54imbi1i 339 . . . 4 (((𝑥𝐵𝜑) → (𝑥𝐴𝜑)) ↔ (¬ (𝑥𝐵 ∧ ¬ 𝜑) → (𝑥𝐴𝜑)))
62, 3, 53bitr4i 292 . . 3 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
76abbii 2736 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
81, 7eqtri 2643 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  {cab 2607  ifcif 4060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-if 4061
This theorem is referenced by:  iftrue  4066  nfifd  4088
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