Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rblem7 Structured version   Visualization version   GIF version

Theorem rblem7 1686
 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
rblem7.1 ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))
Assertion
Ref Expression
rblem7 𝜓𝜑)

Proof of Theorem rblem7
StepHypRef Expression
1 rblem7.1 . 2 ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))
2 rb-ax3 1677 . . 3 (¬ ¬ (¬ 𝜓𝜑) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)))
3 rblem5 1684 . . 3 (¬ (¬ ¬ (¬ 𝜓𝜑) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))) ∨ (¬ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ (¬ 𝜓𝜑)))
42, 3anmp 1674 . 2 (¬ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ (¬ 𝜓𝜑))
51, 4anmp 1674 1 𝜓𝜑)
 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:  re2luk1  1688  re2luk2  1689  re2luk3  1690
 Copyright terms: Public domain W3C validator