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Theorem dfifp7 1050
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
Assertion
Ref Expression
dfifp7 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))

Proof of Theorem dfifp7
StepHypRef Expression
1 orcom 856 . 2 (((𝜑𝜓) ∨ ¬ (𝜒𝜑)) ↔ (¬ (𝜒𝜑) ∨ (𝜑𝜓)))
2 dfifp6 1049 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))
3 imor 839 . 2 (((𝜒𝜑) → (𝜑𝜓)) ↔ (¬ (𝜒𝜑) ∨ (𝜑𝜓)))
41, 2, 33bitr4i 295 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833  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: (None)
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