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Theorem dfif2 4038
 Description: An alternate definition of the conditional operator df-if 4037 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 4037 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 df-or 384 . . . 4 (((𝑥𝐵 ∧ ¬ 𝜑) ∨ (𝑥𝐴𝜑)) ↔ (¬ (𝑥𝐵 ∧ ¬ 𝜑) → (𝑥𝐴𝜑)))
3 orcom 401 . . . 4 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐵 ∧ ¬ 𝜑) ∨ (𝑥𝐴𝜑)))
4 iman 439 . . . . 5 ((𝑥𝐵𝜑) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝜑))
54imbi1i 338 . . . 4 (((𝑥𝐵𝜑) → (𝑥𝐴𝜑)) ↔ (¬ (𝑥𝐵 ∧ ¬ 𝜑) → (𝑥𝐴𝜑)))
62, 3, 53bitr4i 291 . . 3 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
76abbii 2726 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
81, 7eqtri 2632 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037 This theorem is referenced by:  iftrue  4042  nfifd  4064
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