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Theorem rblem6 1527
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 ¬ (¬ (¬ φ ψ) ¬ (¬ ψ φ))
Assertion
Ref Expression
rblem6 φ ψ)

Proof of Theorem rblem6
StepHypRef Expression
1 rblem6.1 . 2 ¬ (¬ (¬ φ ψ) ¬ (¬ ψ φ))
2 rb-ax4 1520 . . . . . . 7 (¬ (¬ (¬ φ ψ) ¬ (¬ φ ψ)) ¬ (¬ φ ψ))
3 rb-ax3 1519 . . . . . . 7 (¬ ¬ (¬ φ ψ) (¬ (¬ φ ψ) ¬ (¬ φ ψ)))
42, 3rbsyl 1521 . . . . . 6 (¬ ¬ (¬ φ ψ) ¬ (¬ φ ψ))
5 rb-ax2 1518 . . . . . 6 (¬ (¬ ¬ (¬ φ ψ) ¬ (¬ φ ψ)) (¬ (¬ φ ψ) ¬ ¬ (¬ φ ψ)))
64, 5anmp 1516 . . . . 5 (¬ (¬ φ ψ) ¬ ¬ (¬ φ ψ))
7 rblem3 1524 . . . . 5 (¬ (¬ (¬ φ ψ) ¬ ¬ (¬ φ ψ)) ((¬ (¬ φ ψ) ¬ (¬ ψ φ)) ¬ ¬ (¬ φ ψ)))
86, 7anmp 1516 . . . 4 ((¬ (¬ φ ψ) ¬ (¬ ψ φ)) ¬ ¬ (¬ φ ψ))
9 rb-ax2 1518 . . . 4 (¬ ((¬ (¬ φ ψ) ¬ (¬ ψ φ)) ¬ ¬ (¬ φ ψ)) (¬ ¬ (¬ φ ψ) (¬ (¬ φ ψ) ¬ (¬ ψ φ))))
108, 9anmp 1516 . . 3 (¬ ¬ (¬ φ ψ) (¬ (¬ φ ψ) ¬ (¬ ψ φ)))
11 rblem5 1526 . . 3 (¬ (¬ ¬ (¬ φ ψ) (¬ (¬ φ ψ) ¬ (¬ ψ φ))) (¬ ¬ (¬ (¬ φ ψ) ¬ (¬ ψ φ)) φ ψ)))
1210, 11anmp 1516 . 2 (¬ ¬ (¬ (¬ φ ψ) ¬ (¬ ψ φ)) φ ψ))
131, 12anmp 1516 1 φ ψ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   wo 357
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 177  df-or 359  df-an 360
This theorem is referenced by:  re1axmp  1529  re2luk1  1530  re2luk2  1531
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