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Theorem ifptru 1074
Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4536. This is essentially dedlema 1045. (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 360 . 2 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2 orc 867 . . . 4 (𝜑 → (𝜑𝜒))
32biantrud 531 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜑𝜒))))
4 dfifp3 1065 . . 3 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
53, 4bitr4di 289 . 2 (𝜑 → ((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜒)))
61, 5bitr2d 280 1 (𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  if-wif 1062
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 207  df-an 396  df-or 848  df-ifp 1063
This theorem is referenced by:  ifpfal  1075  ifpid  1076  elimh  1082  dedt  1083  axprlem3  5430  axpr  5432  axprlem3OLD  5433  axprlem4OLD  5434  wlkl1loop  29670  lfgrwlkprop  29719  eupth2lem3lem3  30258  satfv1lem  35346  wl-3xortru  37453  sn-axprlem3  42234
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