Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rp-fakeoranass Structured version   Visualization version   GIF version

Theorem rp-fakeoranass 37379
 Description: A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakeoranass ((𝜑𝜒) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))

Proof of Theorem rp-fakeoranass
StepHypRef Expression
1 rp-fakeanorass 37378 . 2 ((𝜑𝜒) ↔ (((𝜒𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓𝜑))))
2 bicom 212 . . 3 ((((𝜒𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓𝜑))) ↔ ((𝜒 ∧ (𝜓𝜑)) ↔ ((𝜒𝜓) ∨ 𝜑)))
3 ancom 466 . . . . 5 ((𝜒 ∧ (𝜓𝜑)) ↔ ((𝜓𝜑) ∧ 𝜒))
4 orcom 402 . . . . . 6 ((𝜓𝜑) ↔ (𝜑𝜓))
54anbi1i 730 . . . . 5 (((𝜓𝜑) ∧ 𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
63, 5bitri 264 . . . 4 ((𝜒 ∧ (𝜓𝜑)) ↔ ((𝜑𝜓) ∧ 𝜒))
7 orcom 402 . . . . 5 (((𝜒𝜓) ∨ 𝜑) ↔ (𝜑 ∨ (𝜒𝜓)))
8 ancom 466 . . . . . 6 ((𝜒𝜓) ↔ (𝜓𝜒))
98orbi2i 541 . . . . 5 ((𝜑 ∨ (𝜒𝜓)) ↔ (𝜑 ∨ (𝜓𝜒)))
107, 9bitri 264 . . . 4 (((𝜒𝜓) ∨ 𝜑) ↔ (𝜑 ∨ (𝜓𝜒)))
116, 10bibi12i 329 . . 3 (((𝜒 ∧ (𝜓𝜑)) ↔ ((𝜒𝜓) ∨ 𝜑)) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))
122, 11bitri 264 . 2 ((((𝜒𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓𝜑))) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))
131, 12bitri 264 1 ((𝜑𝜒) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384 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 385  df-an 386 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator