Theorem dfif2 4530

Description: An alternate definition of the conditional operator df-if 4529 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 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfif2

Step	Hyp	Ref	Expression
1		df-if 4529	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
2		df-or 845	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		orcom 867	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		iman 401	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))
5	4	imbi1i 349	. . . 4 ⊢ (((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)))
6	2, 3, 5	3bitr4i 303	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)))
7	6	abbii 2801	. 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}
8	1, 7	eqtri 2759	1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105  {cab 2708  ifcif 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-if 4529

This theorem is referenced by:  iftrue  4534  nfifd  4557