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Theorem ifptru 1043
Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4125. This is essentially dedlema 1014. (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 349 . 2 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2 orc 399 . . . 4 (𝜑 → (𝜑𝜒))
32biantrud 527 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜑𝜒))))
4 dfifp3 1035 . . 3 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
53, 4syl6bbr 278 . 2 (𝜑 → ((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜒)))
61, 5bitr2d 269 1 (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  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:  ifpfal  1044  ifpid  1045  elimh  1050  dedt  1051  wlkl1loop  26590  lfgrwlkprop  26640  eupth2lem3lem3  27208
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