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Theorem ifpbi123d 1077
Description: Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)
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, 2imbi12d 345 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜏𝜂)))
4 ifpbi123d.3 . . . 4 (𝜑 → (𝜃𝜁))
51, 4orbi12d 916 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜏𝜁)))
63, 5anbi12d 631 . 2 (𝜑 → (((𝜓𝜒) ∧ (𝜓𝜃)) ↔ ((𝜏𝜂) ∧ (𝜏𝜁))))
7 dfifp3 1063 . 2 (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓𝜒) ∧ (𝜓𝜃)))
8 dfifp3 1063 . 2 (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏𝜂) ∧ (𝜏𝜁)))
96, 7, 83bitr4g 314 1 (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
Colors of variables: wff setvar class
Syntax hints:  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:  ifpbi23d  1079  wkslem1  27974  wkslem2  27975  iswlk  27977  wlkres  28038  redwlk  28040  wlkp1lem8  28048  crctcshwlkn0lem4  28178  crctcshwlkn0lem5  28179  crctcshwlkn0lem6  28180  1wlkdlem4  28504  pfxwlk  33085  satfv1fvfmla1  33385  ifpbi123  41097
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