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Theorem ifptru 1072
 Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4427. This is essentially dedlema 1042. (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 365 . 2 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2 orc 865 . . . 4 (𝜑 → (𝜑𝜒))
32biantrud 536 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜑𝜒))))
4 dfifp3 1062 . . 3 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
53, 4bitr4di 293 . 2 (𝜑 → ((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜒)))
61, 5bitr2d 283 1 (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845  if-wif 1059 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 210  df-an 401  df-or 846  df-ifp 1060 This theorem is referenced by:  ifpfal  1073  ifpid  1074  elimh  1081  dedt  1082  axprlem3  5295  axprlem4  5296  wlkl1loop  27519  lfgrwlkprop  27569  eupth2lem3lem3  28107  satfv1lem  32833  wl-3xortru  35161  sn-axprlem3  39693
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