Theorem ifpim1g 41006

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

Assertion

Ref	Expression
ifpim1g	⊢ ((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓 → 𝜑) ∨ (𝜃 → 𝜒)) ∧ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜃))))

Proof of Theorem ifpim1g

Step	Hyp	Ref	Expression
1		ifpim123g 41005	. 2 ⊢ ((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜒)) ∧ ((𝜓 → 𝜑) ∨ (𝜃 → 𝜒))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜃 → 𝜃)))))
2		id 22	. . . . . 6 ⊢ (𝜒 → 𝜒)
3	2	olci 862	. . . . 5 ⊢ ((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜒))
4	3	biantrur 530	. . . 4 ⊢ (((𝜓 → 𝜑) ∨ (𝜃 → 𝜒)) ↔ (((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜒)) ∧ ((𝜓 → 𝜑) ∨ (𝜃 → 𝜒))))
5	4	bicomi 223	. . 3 ⊢ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜒)) ∧ ((𝜓 → 𝜑) ∨ (𝜃 → 𝜒))) ↔ ((𝜓 → 𝜑) ∨ (𝜃 → 𝜒)))
6		id 22	. . . . . 6 ⊢ (𝜃 → 𝜃)
7	6	olci 862	. . . . 5 ⊢ ((¬ 𝜓 → 𝜑) ∨ (𝜃 → 𝜃))
8	7	biantru 529	. . . 4 ⊢ (((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)) ↔ (((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜃 → 𝜃))))
9	8	bicomi 223	. . 3 ⊢ ((((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜃 → 𝜃))) ↔ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)))
10	5, 9	anbi12i 626	. 2 ⊢ (((((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜒)) ∧ ((𝜓 → 𝜑) ∨ (𝜃 → 𝜒))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 → 𝜃)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜃 → 𝜃)))) ↔ (((𝜓 → 𝜑) ∨ (𝜃 → 𝜒)) ∧ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜃))))
11	1, 10	bitri 274	1 ⊢ ((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓 → 𝜑) ∨ (𝜃 → 𝜒)) ∧ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜃))))

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:  ifp1bi  41007