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Theorem 1fpid3 1049
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 1033 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 1fpid3.1 . . 3 ((𝜑𝜓) → 𝜒)
3 simpr 476 . . 3 ((¬ 𝜑𝜒) → 𝜒)
42, 3jaoi 393 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) → 𝜒)
51, 4sylbi 207 1 (if-(𝜑, 𝜓, 𝜒) → 𝜒)
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:  ifpsnprss  26574
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