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Theorem casesifp 1046
 Description: Version of cases 1011 expressed using if-. Case disjunction according to the value of 𝜑. One can see this as a proof that the two hypotheses characterize the conditional operator for propositions. For the converses, see ifptru 1043 and ifpfal 1044. (Contributed by BJ, 20-Sep-2019.)
Hypotheses
Ref Expression
casesifp.1 (𝜑 → (𝜓𝜒))
casesifp.2 𝜑 → (𝜓𝜃))
Assertion
Ref Expression
casesifp (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))

Proof of Theorem casesifp
StepHypRef Expression
1 casesifp.1 . . 3 (𝜑 → (𝜓𝜒))
2 casesifp.2 . . 3 𝜑 → (𝜓𝜃))
31, 2cases 1011 . 2 (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
4 df-ifp 1033 . 2 (if-(𝜑, 𝜒, 𝜃) ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
53, 4bitr4i 267 1 (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383  if-wif 1032 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 1033 This theorem is referenced by:  hadifp  1584  cadifp  1597
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