Theorem dfif2 3665

Description: An alternate definition of the conditional operator df-if 3664 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 3664	. 2 ⊢ if(φ, A, B) = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}
2		df-or 359	. . . 4 ⊢ (((x ∈ B ∧ ¬ φ) ∨ (x ∈ A ∧ φ)) ↔ (¬ (x ∈ B ∧ ¬ φ) → (x ∈ A ∧ φ)))
3		orcom 376	. . . 4 ⊢ (((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ)) ↔ ((x ∈ B ∧ ¬ φ) ∨ (x ∈ A ∧ φ)))
4		iman 413	. . . . 5 ⊢ ((x ∈ B → φ) ↔ ¬ (x ∈ B ∧ ¬ φ))
5	4	imbi1i 315	. . . 4 ⊢ (((x ∈ B → φ) → (x ∈ A ∧ φ)) ↔ (¬ (x ∈ B ∧ ¬ φ) → (x ∈ A ∧ φ)))
6	2, 3, 5	3bitr4i 268	. . 3 ⊢ (((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ)) ↔ ((x ∈ B → φ) → (x ∈ A ∧ φ)))
7	6	abbii 2466	. 2 ⊢ {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))} = {x ∣ ((x ∈ B → φ) → (x ∈ A ∧ φ))}
8	1, 7	eqtri 2373	1 ⊢ if(φ, A, B) = {x ∣ ((x ∈ B → φ) → (x ∈ A ∧ φ))}

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-if 3664

This theorem is referenced by:  iftrue  3669  nfifd  3686