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Theorem dfif2 4467
Description: An alternate definition of the conditional operator df-if 4466 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 4466 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 df-or 842 . . . 4 (((𝑥𝐵 ∧ ¬ 𝜑) ∨ (𝑥𝐴𝜑)) ↔ (¬ (𝑥𝐵 ∧ ¬ 𝜑) → (𝑥𝐴𝜑)))
3 orcom 864 . . . 4 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐵 ∧ ¬ 𝜑) ∨ (𝑥𝐴𝜑)))
4 iman 402 . . . . 5 ((𝑥𝐵𝜑) ↔ ¬ (𝑥𝐵 ∧ ¬ 𝜑))
54imbi1i 351 . . . 4 (((𝑥𝐵𝜑) → (𝑥𝐴𝜑)) ↔ (¬ (𝑥𝐵 ∧ ¬ 𝜑) → (𝑥𝐴𝜑)))
62, 3, 53bitr4i 304 . . 3 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑)))
76abbii 2886 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
81, 7eqtri 2844 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  {cab 2799  ifcif 4465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-sb 2061  df-clab 2800  df-cleq 2814  df-if 4466
This theorem is referenced by:  iftrue  4471  nfifd  4493
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