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Theorem 1fpid3 1060
 Description: The value of the conditional operator for propositions is its third argument if the first and second argument imply the third argument. (Contributed by AV, 4-Apr-2021.)
Hypothesis
Ref Expression
1fpid3.1 ((𝜑𝜓) → 𝜒)
Assertion
Ref Expression
1fpid3 (if-(𝜑, 𝜓, 𝜒) → 𝜒)

Proof of Theorem 1fpid3
StepHypRef Expression
1 df-ifp 1044 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 1fpid3.1 . . 3 ((𝜑𝜓) → 𝜒)
3 simpr 477 . . 3 ((¬ 𝜑𝜒) → 𝜒)
42, 3jaoi 843 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) → 𝜒)
51, 4sylbi 209 1 (if-(𝜑, 𝜓, 𝜒) → 𝜒)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ 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:  ifpsnprss  27101
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