Theorem dfif2 4494

Description: An alternate definition of the conditional operator df-if 4493 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 4493	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
2		df-or 861	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		orcom 883	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		iman 406	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))
5	4	imbi1i 352	. . . 4 ⊢ (((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)))
6	2, 3, 5	3bitr4i 306	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)))
7	6	abbii 2836	. 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}
8	1, 7	eqtri 2792	1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  {cab 2747  ifcif 4492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-if 4493

This theorem is referenced by:  iftrue  4498  nfifd  4522