 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifptru Structured version   Visualization version   GIF version

Theorem ifptru 1061
 Description: Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4236. This is essentially dedlema 1032. (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 529 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜑𝜒))))
4 dfifp3 1053 . . 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 1050 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 1051 This theorem is referenced by:  ifpfal  1062  ifpid  1063  elimh  1068  dedt  1069  wlkl1loop  26765  lfgrwlkprop  26815  eupth2lem3lem3  27403
 Copyright terms: Public domain W3C validator