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Theorem ifptru 1075
Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4531. This is essentially dedlema 1046. (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 868 . . . 4 (𝜑 → (𝜑𝜒))
32biantrud 531 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜑𝜒))))
4 dfifp3 1066 . . 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 848  if-wif 1063
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 849  df-ifp 1064
This theorem is referenced by:  ifpfal  1076  ifpid  1077  elimh  1083  dedt  1084  axprlem3  5425  axpr  5427  axprlem3OLD  5428  axprlem4OLD  5429  wlkl1loop  29656  lfgrwlkprop  29705  eupth2lem3lem3  30249  satfv1lem  35367  wl-3xortru  37472  sn-axprlem3  42256
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