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Theorem ifpnOLD 1070
 Description: Obsolete version of ifpn 1069 as of 5-May-2024.. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ifpnOLD (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))

Proof of Theorem ifpnOLD
StepHypRef Expression
1 notnotb 318 . . . 4 (𝜑 ↔ ¬ ¬ 𝜑)
21imbi1i 353 . . 3 ((𝜑𝜓) ↔ (¬ ¬ 𝜑𝜓))
32anbi2ci 627 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜒) ∧ (¬ ¬ 𝜑𝜓)))
4 dfifp2 1060 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
5 dfifp2 1060 . 2 (if-(¬ 𝜑, 𝜒, 𝜓) ↔ ((¬ 𝜑𝜒) ∧ (¬ ¬ 𝜑𝜓)))
63, 4, 53bitr4i 306 1 (if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  if-wif 1058 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 210  df-an 400  df-or 845  df-ifp 1059 This theorem is referenced by: (None)
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