MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rblem6 Structured version   Visualization version   GIF version

Theorem rblem6 1685
Description: Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
rblem6.1 ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))
Assertion
Ref Expression
rblem6 𝜑𝜓)

Proof of Theorem rblem6
StepHypRef Expression
1 rblem6.1 . 2 ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))
2 rb-ax4 1678 . . . . . . 7 (¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜑𝜓)) ∨ ¬ (¬ 𝜑𝜓))
3 rb-ax3 1677 . . . . . . 7 (¬ ¬ (¬ 𝜑𝜓) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜑𝜓)))
42, 3rbsyl 1679 . . . . . 6 (¬ ¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜑𝜓))
5 rb-ax2 1676 . . . . . 6 (¬ (¬ ¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜑𝜓)) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ ¬ (¬ 𝜑𝜓)))
64, 5anmp 1674 . . . . 5 (¬ (¬ 𝜑𝜓) ∨ ¬ ¬ (¬ 𝜑𝜓))
7 rblem3 1682 . . . . 5 (¬ (¬ (¬ 𝜑𝜓) ∨ ¬ ¬ (¬ 𝜑𝜓)) ∨ ((¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ ¬ ¬ (¬ 𝜑𝜓)))
86, 7anmp 1674 . . . 4 ((¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ ¬ ¬ (¬ 𝜑𝜓))
9 rb-ax2 1676 . . . 4 (¬ ((¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ ¬ ¬ (¬ 𝜑𝜓)) ∨ (¬ ¬ (¬ 𝜑𝜓) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))))
108, 9anmp 1674 . . 3 (¬ ¬ (¬ 𝜑𝜓) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)))
11 rblem5 1684 . . 3 (¬ (¬ ¬ (¬ 𝜑𝜓) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))) ∨ (¬ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ (¬ 𝜑𝜓)))
1210, 11anmp 1674 . 2 (¬ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ (¬ 𝜑𝜓))
131, 12anmp 1674 1 𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383
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:  re1axmp  1687  re2luk1  1688  re2luk2  1689
  Copyright terms: Public domain W3C validator