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Theorem ifpimpda 1059
 Description: Separation of the values of the conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 27-Feb-2021.)
Hypotheses
Ref Expression
ifpimpda.1 ((𝜑𝜓) → 𝜒)
ifpimpda.2 ((𝜑 ∧ ¬ 𝜓) → 𝜃)
Assertion
Ref Expression
ifpimpda (𝜑 → if-(𝜓, 𝜒, 𝜃))

Proof of Theorem ifpimpda
StepHypRef Expression
1 ifpimpda.1 . . 3 ((𝜑𝜓) → 𝜒)
21ex 405 . 2 (𝜑 → (𝜓𝜒))
3 ifpimpda.2 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝜃)
43ex 405 . 2 (𝜑 → (¬ 𝜓𝜃))
5 dfifp2 1045 . 2 (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓𝜒) ∧ (¬ 𝜓𝜃)))
62, 4, 5sylanbrc 575 1 (𝜑 → if-(𝜓, 𝜒, 𝜃))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387  if-wif 1043 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 199  df-an 388  df-or 834  df-ifp 1044 This theorem is referenced by:  ifpprsnss  4574  wlkp1lem8  27168  1wlkdlem4  27669
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