Theorem ifpbi1b 41110

Description: When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.)

Assertion

Ref	Expression
ifpbi1b	⊢ (if-(𝜑, 𝜒, 𝜒) ↔ if-(𝜓, 𝜒, 𝜒))

Proof of Theorem ifpbi1b

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝜒 → 𝜒)
2	1	olci 863	. . . 4 ⊢ ((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜒))
3	1	olci 863	. . . 4 ⊢ ((𝜓 → 𝜑) ∨ (𝜒 → 𝜒))
4	2, 3	pm3.2i 471	. . 3 ⊢ (((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜒)) ∧ ((𝜓 → 𝜑) ∨ (𝜒 → 𝜒)))
5	1	olci 863	. . . 4 ⊢ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜒))
6	1	olci 863	. . . 4 ⊢ ((¬ 𝜓 → 𝜑) ∨ (𝜒 → 𝜒))
7	5, 6	pm3.2i 471	. . 3 ⊢ (((𝜑 → 𝜓) ∨ (𝜒 → 𝜒)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜒 → 𝜒)))
8		ifpim123g 41107	. . 3 ⊢ ((if-(𝜑, 𝜒, 𝜒) → if-(𝜓, 𝜒, 𝜒)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 → 𝜒)) ∧ ((𝜓 → 𝜑) ∨ (𝜒 → 𝜒))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 → 𝜒)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜒 → 𝜒)))))
9	4, 7, 8	mpbir2an 708	. 2 ⊢ (if-(𝜑, 𝜒, 𝜒) → if-(𝜓, 𝜒, 𝜒))
10	1	olci 863	. . . 4 ⊢ ((𝜓 → ¬ 𝜑) ∨ (𝜒 → 𝜒))
11	10, 5	pm3.2i 471	. . 3 ⊢ (((𝜓 → ¬ 𝜑) ∨ (𝜒 → 𝜒)) ∧ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜒)))
12	1	olci 863	. . . 4 ⊢ ((¬ 𝜑 → 𝜓) ∨ (𝜒 → 𝜒))
13	3, 12	pm3.2i 471	. . 3 ⊢ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜒)) ∧ ((¬ 𝜑 → 𝜓) ∨ (𝜒 → 𝜒)))
14		ifpim123g 41107	. . 3 ⊢ ((if-(𝜓, 𝜒, 𝜒) → if-(𝜑, 𝜒, 𝜒)) ↔ ((((𝜓 → ¬ 𝜑) ∨ (𝜒 → 𝜒)) ∧ ((𝜑 → 𝜓) ∨ (𝜒 → 𝜒))) ∧ (((𝜓 → 𝜑) ∨ (𝜒 → 𝜒)) ∧ ((¬ 𝜑 → 𝜓) ∨ (𝜒 → 𝜒)))))
15	11, 13, 14	mpbir2an 708	. 2 ⊢ (if-(𝜓, 𝜒, 𝜒) → if-(𝜑, 𝜒, 𝜒))
16	9, 15	impbii 208	1 ⊢ (if-(𝜑, 𝜒, 𝜒) ↔ if-(𝜓, 𝜒, 𝜒))

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)