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Theorem rblem6 1756
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 1749 . . . . . . 7 (¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜑𝜓)) ∨ ¬ (¬ 𝜑𝜓))
3 rb-ax3 1748 . . . . . . 7 (¬ ¬ (¬ 𝜑𝜓) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜑𝜓)))
42, 3rbsyl 1750 . . . . . 6 (¬ ¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜑𝜓))
5 rb-ax2 1747 . . . . . 6 (¬ (¬ ¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜑𝜓)) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ ¬ (¬ 𝜑𝜓)))
64, 5anmp 1745 . . . . 5 (¬ (¬ 𝜑𝜓) ∨ ¬ ¬ (¬ 𝜑𝜓))
7 rblem3 1753 . . . . 5 (¬ (¬ (¬ 𝜑𝜓) ∨ ¬ ¬ (¬ 𝜑𝜓)) ∨ ((¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ ¬ ¬ (¬ 𝜑𝜓)))
86, 7anmp 1745 . . . 4 ((¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ ¬ ¬ (¬ 𝜑𝜓))
9 rb-ax2 1747 . . . 4 (¬ ((¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ ¬ ¬ (¬ 𝜑𝜓)) ∨ (¬ ¬ (¬ 𝜑𝜓) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))))
108, 9anmp 1745 . . 3 (¬ ¬ (¬ 𝜑𝜓) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)))
11 rblem5 1755 . . 3 (¬ (¬ ¬ (¬ 𝜑𝜓) ∨ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑))) ∨ (¬ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ (¬ 𝜑𝜓)))
1210, 11anmp 1745 . 2 (¬ ¬ (¬ (¬ 𝜑𝜓) ∨ ¬ (¬ 𝜓𝜑)) ∨ (¬ 𝜑𝜓))
131, 12anmp 1745 1 𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 843
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 208  df-an 397  df-or 844
This theorem is referenced by:  re1axmp  1758  re2luk1  1759  re2luk2  1760
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