Theorem dfif2 4491

Description: An alternate definition of the conditional operator df-if 4490 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 4490	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
2		df-or 862	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)))
3		orcom 884	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		iman 407	. . . . 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 2832	. 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}
8	1, 7	eqtri 2788	1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  {cab 2743  ifcif 4489

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-if 4490

This theorem is used by:  iftrue  4495  nfifd  4519