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Theorem ifpbi1b 38165
 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
StepHypRef Expression
1 id 22 . . . . 5 (𝜒𝜒)
21olci 405 . . . 4 ((𝜑 → ¬ 𝜓) ∨ (𝜒𝜒))
31olci 405 . . . 4 ((𝜓𝜑) ∨ (𝜒𝜒))
42, 3pm3.2i 470 . . 3 (((𝜑 → ¬ 𝜓) ∨ (𝜒𝜒)) ∧ ((𝜓𝜑) ∨ (𝜒𝜒)))
51olci 405 . . . 4 ((𝜑𝜓) ∨ (𝜒𝜒))
61olci 405 . . . 4 ((¬ 𝜓𝜑) ∨ (𝜒𝜒))
75, 6pm3.2i 470 . . 3 (((𝜑𝜓) ∨ (𝜒𝜒)) ∧ ((¬ 𝜓𝜑) ∨ (𝜒𝜒)))
8 ifpim123g 38162 . . 3 ((if-(𝜑, 𝜒, 𝜒) → if-(𝜓, 𝜒, 𝜒)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜒)) ∧ ((𝜓𝜑) ∨ (𝜒𝜒))) ∧ (((𝜑𝜓) ∨ (𝜒𝜒)) ∧ ((¬ 𝜓𝜑) ∨ (𝜒𝜒)))))
94, 7, 8mpbir2an 975 . 2 (if-(𝜑, 𝜒, 𝜒) → if-(𝜓, 𝜒, 𝜒))
101olci 405 . . . 4 ((𝜓 → ¬ 𝜑) ∨ (𝜒𝜒))
1110, 5pm3.2i 470 . . 3 (((𝜓 → ¬ 𝜑) ∨ (𝜒𝜒)) ∧ ((𝜑𝜓) ∨ (𝜒𝜒)))
121olci 405 . . . 4 ((¬ 𝜑𝜓) ∨ (𝜒𝜒))
133, 12pm3.2i 470 . . 3 (((𝜓𝜑) ∨ (𝜒𝜒)) ∧ ((¬ 𝜑𝜓) ∨ (𝜒𝜒)))
14 ifpim123g 38162 . . 3 ((if-(𝜓, 𝜒, 𝜒) → if-(𝜑, 𝜒, 𝜒)) ↔ ((((𝜓 → ¬ 𝜑) ∨ (𝜒𝜒)) ∧ ((𝜑𝜓) ∨ (𝜒𝜒))) ∧ (((𝜓𝜑) ∨ (𝜒𝜒)) ∧ ((¬ 𝜑𝜓) ∨ (𝜒𝜒)))))
1511, 13, 14mpbir2an 975 . 2 (if-(𝜓, 𝜒, 𝜒) → if-(𝜑, 𝜒, 𝜒))
169, 15impbii 199 1 (if-(𝜑, 𝜒, 𝜒) ↔ if-(𝜓, 𝜒, 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383  if-wif 1032 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 197  df-or 384  df-an 385  df-ifp 1033 This theorem is referenced by: (None)
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