Theorem ifpor123g 41115

Description: Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)

Assertion

Ref	Expression
ifpor123g	⊢ ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 ∨ 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 ∨ 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜂)))))

Proof of Theorem ifpor123g

Step	Hyp	Ref	Expression
1		df-or 845	. . . 4 ⊢ ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ (¬ if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)))
2		ifpnot23 41085	. . . . 5 ⊢ (¬ if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜑, ¬ 𝜒, ¬ 𝜏))
3	2	imbi1i 350	. . . 4 ⊢ ((¬ if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ (if-(𝜑, ¬ 𝜒, ¬ 𝜏) → if-(𝜓, 𝜃, 𝜂)))
4	1, 3	bitri 274	. . 3 ⊢ ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ (if-(𝜑, ¬ 𝜒, ¬ 𝜏) → if-(𝜓, 𝜃, 𝜂)))
5		ifpim123g 41107	. . 3 ⊢ ((if-(𝜑, ¬ 𝜒, ¬ 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒 → 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (¬ 𝜒 → 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜂)))))
6	4, 5	bitri 274	. 2 ⊢ ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒 → 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (¬ 𝜒 → 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜂)))))
7		pm4.64 846	. . . . 5 ⊢ ((¬ 𝜒 → 𝜃) ↔ (𝜒 ∨ 𝜃))
8	7	orbi2i 910	. . . 4 ⊢ (((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒 → 𝜃)) ↔ ((𝜑 → ¬ 𝜓) ∨ (𝜒 ∨ 𝜃)))
9		pm4.64 846	. . . . 5 ⊢ ((¬ 𝜏 → 𝜃) ↔ (𝜏 ∨ 𝜃))
10	9	orbi2i 910	. . . 4 ⊢ (((𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜃)) ↔ ((𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜃)))
11	8, 10	anbi12i 627	. . 3 ⊢ ((((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒 → 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜃))) ↔ (((𝜑 → ¬ 𝜓) ∨ (𝜒 ∨ 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜃))))
12		pm4.64 846	. . . . 5 ⊢ ((¬ 𝜒 → 𝜂) ↔ (𝜒 ∨ 𝜂))
13	12	orbi2i 910	. . . 4 ⊢ (((𝜑 → 𝜓) ∨ (¬ 𝜒 → 𝜂)) ↔ ((𝜑 → 𝜓) ∨ (𝜒 ∨ 𝜂)))
14		pm4.64 846	. . . . 5 ⊢ ((¬ 𝜏 → 𝜂) ↔ (𝜏 ∨ 𝜂))
15	14	orbi2i 910	. . . 4 ⊢ (((¬ 𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜂)) ↔ ((¬ 𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜂)))
16	13, 15	anbi12i 627	. . 3 ⊢ ((((𝜑 → 𝜓) ∨ (¬ 𝜒 → 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜂))) ↔ (((𝜑 → 𝜓) ∨ (𝜒 ∨ 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜂))))
17	11, 16	anbi12i 627	. 2 ⊢ (((((𝜑 → ¬ 𝜓) ∨ (¬ 𝜒 → 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (¬ 𝜒 → 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (¬ 𝜏 → 𝜂)))) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 ∨ 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 ∨ 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜂)))))
18	6, 17	bitri 274	1 ⊢ ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 ∨ 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 ∨ 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜂)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  if-wif 1060

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 397  df-or 845  df-ifp 1061

This theorem is referenced by: (None)