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Theorem ifpimpda 1027
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 450 . 2 (𝜑 → (𝜓𝜒))
3 ifpimpda.2 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝜃)
43ex 450 . 2 (𝜑 → (¬ 𝜓𝜃))
5 dfifp2 1013 . 2 (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓𝜒) ∧ (¬ 𝜓𝜃)))
62, 4, 5sylanbrc 697 1 (𝜑 → if-(𝜓, 𝜒, 𝜃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  if-wif 1011
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 385  df-an 386  df-ifp 1012
This theorem is referenced by:  ifpprsnss  4290  wlkp1lem8  26558  1wlkdlem4  26980
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