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Theorem ifptru 1016
 Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4037. This is essentially dedlema 992. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
Assertion
Ref Expression
ifptru (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))

Proof of Theorem ifptru
StepHypRef Expression
1 biimt 348 . 2 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2 orc 398 . . . 4 (𝜑 → (𝜑𝜒))
32biantrud 526 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜑𝜒))))
4 dfifp3 1008 . . 3 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
53, 4syl6bbr 276 . 2 (𝜑 → ((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜒)))
61, 5bitr2d 267 1 (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 194   ∨ wo 381   ∧ wa 382  if-wif 1005 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 195  df-or 383  df-an 384  df-ifp 1006 This theorem is referenced by:  ifpfal  1017  ifpid  1018  elimh  1023  dedt  1024  1wlkl1loop  40840  lfgrwlkprop  40894  eupth2lem3lem3  41396
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