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Theorem ifpbi23d 1077
Description: Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.)
Hypotheses
Ref Expression
ifpbi23d.1 (𝜑 → (𝜒𝜂))
ifpbi23d.2 (𝜑 → (𝜃𝜁))
Assertion
Ref Expression
ifpbi23d (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))

Proof of Theorem ifpbi23d
StepHypRef Expression
1 biidd 265 . 2 (𝜑 → (𝜓𝜓))
2 ifpbi23d.1 . 2 (𝜑 → (𝜒𝜂))
3 ifpbi23d.2 . 2 (𝜑 → (𝜃𝜁))
41, 2, 3ifpbi123d 1075 1 (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  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:  wksfval  27399  subgrwlk  32492  satfv1fvfmla1  32783  bj-ififc  34028  wl-df-3xor  34885  wl-df3maxtru1  34909
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