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Theorem ifpbi123dOLD 1076
 Description: Obsolete version of ifpbi123d 1075 as of 17-Apr-2024. (Contributed by AV, 30-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ifpbi123d.1 (𝜑 → (𝜓𝜏))
ifpbi123d.2 (𝜑 → (𝜒𝜂))
ifpbi123d.3 (𝜑 → (𝜃𝜁))
Assertion
Ref Expression
ifpbi123dOLD (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Proof of Theorem ifpbi123dOLD
StepHypRef Expression
1 ifpbi123d.1 . . . 4 (𝜑 → (𝜓𝜏))
2 ifpbi123d.2 . . . 4 (𝜑 → (𝜒𝜂))
31, 2anbi12d 633 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜏𝜂)))
41notbid 321 . . . 4 (𝜑 → (¬ 𝜓 ↔ ¬ 𝜏))
5 ifpbi123d.3 . . . 4 (𝜑 → (𝜃𝜁))
64, 5anbi12d 633 . . 3 (𝜑 → ((¬ 𝜓𝜃) ↔ (¬ 𝜏𝜁)))
73, 6orbi12d 916 . 2 (𝜑 → (((𝜓𝜒) ∨ (¬ 𝜓𝜃)) ↔ ((𝜏𝜂) ∨ (¬ 𝜏𝜁))))
8 df-ifp 1059 . 2 (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓𝜒) ∨ (¬ 𝜓𝜃)))
9 df-ifp 1059 . 2 (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏𝜂) ∨ (¬ 𝜏𝜁)))
107, 8, 93bitr4g 317 1 (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  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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