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Theorem ifpbi123d 1065
Description: Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.)
Hypotheses
Ref Expression
ifpbi123d.1 (𝜑 → (𝜓𝜏))
ifpbi123d.2 (𝜑 → (𝜒𝜂))
ifpbi123d.3 (𝜑 → (𝜃𝜁))
Assertion
Ref Expression
ifpbi123d (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Proof of Theorem ifpbi123d
StepHypRef Expression
1 ifpbi123d.1 . . . 4 (𝜑 → (𝜓𝜏))
2 ifpbi123d.2 . . . 4 (𝜑 → (𝜒𝜂))
31, 2anbi12d 749 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜏𝜂)))
41notbid 307 . . . 4 (𝜑 → (¬ 𝜓 ↔ ¬ 𝜏))
5 ifpbi123d.3 . . . 4 (𝜑 → (𝜃𝜁))
64, 5anbi12d 749 . . 3 (𝜑 → ((¬ 𝜓𝜃) ↔ (¬ 𝜏𝜁)))
73, 6orbi12d 748 . 2 (𝜑 → (((𝜓𝜒) ∨ (¬ 𝜓𝜃)) ↔ ((𝜏𝜂) ∨ (¬ 𝜏𝜁))))
8 df-ifp 1051 . 2 (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓𝜒) ∨ (¬ 𝜓𝜃)))
9 df-ifp 1051 . 2 (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏𝜂) ∨ (¬ 𝜏𝜁)))
107, 8, 93bitr4g 303 1 (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  if-wif 1050
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 1051
This theorem is referenced by:  wkslem1  26713  wkslem2  26714  wksfval  26715  iswlk  26716  wlkres  26777  redwlk  26779  wlkp1lem8  26787  crctcshwlkn0lem4  26916  crctcshwlkn0lem5  26917  crctcshwlkn0lem6  26918  1wlkdlem4  27292
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