Users' Mathboxes Mathbox for Anthony Hart < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  waj-ax Structured version   Visualization version   GIF version

Theorem waj-ax 32750
Description: A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)
Assertion
Ref Expression
waj-ax ((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓))))

Proof of Theorem waj-ax
StepHypRef Expression
1 nannan 1599 . . 3 ((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
2 simpr 471 . . . . . . . . 9 ((𝜓𝜒) → 𝜒)
32imim2i 16 . . . . . . . 8 ((𝜑 → (𝜓𝜒)) → (𝜑𝜒))
4 pm2.27 42 . . . . . . . . . 10 (𝜑 → ((𝜑𝜒) → 𝜒))
54anim2d 599 . . . . . . . . 9 (𝜑 → ((𝜃 ∧ (𝜑𝜒)) → (𝜃𝜒)))
65expdimp 440 . . . . . . . 8 ((𝜑𝜃) → ((𝜑𝜒) → (𝜃𝜒)))
73, 6syl5com 31 . . . . . . 7 ((𝜑 → (𝜓𝜒)) → ((𝜑𝜃) → (𝜃𝜒)))
87con3d 149 . . . . . 6 ((𝜑 → (𝜓𝜒)) → (¬ (𝜃𝜒) → ¬ (𝜑𝜃)))
9 df-nan 1596 . . . . . 6 ((𝜃𝜒) ↔ ¬ (𝜃𝜒))
10 df-nan 1596 . . . . . 6 ((𝜑𝜃) ↔ ¬ (𝜑𝜃))
118, 9, 103imtr4g 285 . . . . 5 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) → (𝜑𝜃)))
12 nanim 1600 . . . . 5 (((𝜃𝜒) → (𝜑𝜃)) ↔ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))
1311, 12sylib 208 . . . 4 ((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))
14 pm3.21 457 . . . . . . . 8 (𝜓 → (𝜑 → (𝜑𝜓)))
1514adantr 466 . . . . . . 7 ((𝜓𝜒) → (𝜑 → (𝜑𝜓)))
1615com12 32 . . . . . 6 (𝜑 → ((𝜓𝜒) → (𝜑𝜓)))
1716a2i 14 . . . . 5 ((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜑𝜓)))
18 nannan 1599 . . . . 5 ((𝜑 ⊼ (𝜑𝜓)) ↔ (𝜑 → (𝜑𝜓)))
1917, 18sylibr 224 . . . 4 ((𝜑 → (𝜓𝜒)) → (𝜑 ⊼ (𝜑𝜓)))
2013, 19jca 501 . . 3 ((𝜑 → (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓))))
211, 20sylbi 207 . 2 ((𝜑 ⊼ (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓))))
22 nannan 1599 . 2 (((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓)))) ↔ ((𝜑 ⊼ (𝜓𝜒)) → (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ∧ (𝜑 ⊼ (𝜑𝜓)))))
2321, 22mpbir 221 1 ((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wnan 1595
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-an 383  df-nan 1596
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator