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Theorem ifpor 1041
Description: The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1040. (Contributed by BJ, 1-Oct-2019.)
Assertion
Ref Expression
ifpor (if-(𝜑, 𝜓, 𝜒) → (𝜓𝜒))

Proof of Theorem ifpor
StepHypRef Expression
1 df-ifp 1033 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 simpr 476 . . 3 ((𝜑𝜓) → 𝜓)
3 simpr 476 . . 3 ((¬ 𝜑𝜒) → 𝜒)
42, 3orim12i 537 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) → (𝜓𝜒))
51, 4sylbi 207 1 (if-(𝜑, 𝜓, 𝜒) → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  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: (None)
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