Theorem ifpim23g 41000

Description: Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpim23g	⊢ (((𝜑 → 𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ (𝜒 → (𝜑 ∨ 𝜓))))

Proof of Theorem ifpim23g

Step	Hyp	Ref	Expression
1		ifpidg 40996	. 2 ⊢ (((𝜑 → 𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ ((((𝜒 ∧ 𝜓) → (𝜑 → 𝜓)) ∧ ((𝜒 ∧ (𝜑 → 𝜓)) → 𝜓)) ∧ ((¬ 𝜑 → (𝜒 ∨ (𝜑 → 𝜓))) ∧ ((𝜑 → 𝜓) → (𝜒 ∨ ¬ 𝜑)))))
2		dfor2 898	. . . . 5 ⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓))
3	2	imbi2i 335	. . . 4 ⊢ ((𝜒 → (𝜑 ∨ 𝜓)) ↔ (𝜒 → ((𝜑 → 𝜓) → 𝜓)))
4		impexp 450	. . . 4 ⊢ (((𝜒 ∧ (𝜑 → 𝜓)) → 𝜓) ↔ (𝜒 → ((𝜑 → 𝜓) → 𝜓)))
5		ax-1 6	. . . . . 6 ⊢ (𝜓 → (𝜑 → 𝜓))
6	5	adantl 481	. . . . 5 ⊢ ((𝜒 ∧ 𝜓) → (𝜑 → 𝜓))
7	6	biantrur 530	. . . 4 ⊢ (((𝜒 ∧ (𝜑 → 𝜓)) → 𝜓) ↔ (((𝜒 ∧ 𝜓) → (𝜑 → 𝜓)) ∧ ((𝜒 ∧ (𝜑 → 𝜓)) → 𝜓)))
8	3, 4, 7	3bitr2i 298	. . 3 ⊢ ((𝜒 → (𝜑 ∨ 𝜓)) ↔ (((𝜒 ∧ 𝜓) → (𝜑 → 𝜓)) ∧ ((𝜒 ∧ (𝜑 → 𝜓)) → 𝜓)))
9		impexp 450	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒)))
10		imdi 390	. . . . . 6 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
11		imor 849	. . . . . . . 8 ⊢ ((𝜑 → 𝜒) ↔ (¬ 𝜑 ∨ 𝜒))
12		orcom 866	. . . . . . . 8 ⊢ ((¬ 𝜑 ∨ 𝜒) ↔ (𝜒 ∨ ¬ 𝜑))
13	11, 12	bitri 274	. . . . . . 7 ⊢ ((𝜑 → 𝜒) ↔ (𝜒 ∨ ¬ 𝜑))
14	13	imbi2i 335	. . . . . 6 ⊢ (((𝜑 → 𝜓) → (𝜑 → 𝜒)) ↔ ((𝜑 → 𝜓) → (𝜒 ∨ ¬ 𝜑)))
15	10, 14	bitri 274	. . . . 5 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 → 𝜓) → (𝜒 ∨ ¬ 𝜑)))
16	9, 15	bitri 274	. . . 4 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → (𝜒 ∨ ¬ 𝜑)))
17		pm2.21 123	. . . . . 6 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
18	17	olcd 870	. . . . 5 ⊢ (¬ 𝜑 → (𝜒 ∨ (𝜑 → 𝜓)))
19	18	biantrur 530	. . . 4 ⊢ (((𝜑 → 𝜓) → (𝜒 ∨ ¬ 𝜑)) ↔ ((¬ 𝜑 → (𝜒 ∨ (𝜑 → 𝜓))) ∧ ((𝜑 → 𝜓) → (𝜒 ∨ ¬ 𝜑))))
20	16, 19	bitri 274	. . 3 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((¬ 𝜑 → (𝜒 ∨ (𝜑 → 𝜓))) ∧ ((𝜑 → 𝜓) → (𝜒 ∨ ¬ 𝜑))))
21	8, 20	anbi12i 626	. 2 ⊢ (((𝜒 → (𝜑 ∨ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜒)) ↔ ((((𝜒 ∧ 𝜓) → (𝜑 → 𝜓)) ∧ ((𝜒 ∧ (𝜑 → 𝜓)) → 𝜓)) ∧ ((¬ 𝜑 → (𝜒 ∨ (𝜑 → 𝜓))) ∧ ((𝜑 → 𝜓) → (𝜒 ∨ ¬ 𝜑)))))
22		ancom 460	. 2 ⊢ (((𝜒 → (𝜑 ∨ 𝜓)) ∧ ((𝜑 ∧ 𝜓) → 𝜒)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ (𝜒 → (𝜑 ∨ 𝜓))))
23	1, 21, 22	3bitr2i 298	1 ⊢ (((𝜑 → 𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ (𝜒 → (𝜑 ∨ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843  if-wif 1059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by:  ifpim3  41001  ifpim4  41003