Mathbox for Wolf Lammen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-ifp4impr Structured version   Visualization version   GIF version

Theorem wl-ifp4impr 34877
 Description: If one case of an if- condition is a consequence of the other, the expression in dfifp4 1062 can be shortened. (Contributed by Wolf Lammen, 18-Jun-2024.)
Assertion
Ref Expression
wl-ifp4impr ((𝜒𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓)))

Proof of Theorem wl-ifp4impr
StepHypRef Expression
1 wl-ifpimpr 34876 . 2 ((𝜒𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ 𝜒)))
2 pm4.71 561 . . . . 5 ((𝜒𝜓) ↔ (𝜒 ↔ (𝜒𝜓)))
32biimpi 219 . . . 4 ((𝜒𝜓) → (𝜒 ↔ (𝜒𝜓)))
43orbi2d 913 . . 3 ((𝜒𝜓) → (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜓) ∨ (𝜒𝜓))))
5 andir 1006 . . 3 (((𝜑𝜒) ∧ 𝜓) ↔ ((𝜑𝜓) ∨ (𝜒𝜓)))
64, 5syl6bbr 292 . 2 ((𝜒𝜓) → (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓)))
71, 6bitrd 282 1 ((𝜒𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058 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 400  df-or 845  df-ifp 1059 This theorem is referenced by:  wl-df3-3mintru2  34896
 Copyright terms: Public domain W3C validator