Theorem tbwlem5 1474

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tbwlem5	⊢ (((φ → (ψ → ⊥ )) → ⊥ ) → φ)

Proof of Theorem tbwlem5

Step	Hyp	Ref	Expression
1		tbw-ax2 1466	. . . 4 ⊢ (φ → (ψ → φ))
2		tbw-ax1 1465	. . . 4 ⊢ ((ψ → φ) → ((φ → ⊥ ) → (ψ → ⊥ )))
3	1, 2	tbwsyl 1469	. . 3 ⊢ (φ → ((φ → ⊥ ) → (ψ → ⊥ )))
4		tbwlem1 1470	. . 3 ⊢ ((φ → ((φ → ⊥ ) → (ψ → ⊥ ))) → ((φ → ⊥ ) → (φ → (ψ → ⊥ ))))
5	3, 4	ax-mp 5	. 2 ⊢ ((φ → ⊥ ) → (φ → (ψ → ⊥ )))
6		tbwlem4 1473	. 2 ⊢ (((φ → ⊥ ) → (φ → (ψ → ⊥ ))) → (((φ → (ψ → ⊥ )) → ⊥ ) → φ))
7	5, 6	ax-mp 5	1 ⊢ (((φ → (ψ → ⊥ )) → ⊥ ) → φ)

Syntax hints:   → wi 4   ⊥ wfal 1317

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-tru 1319  df-fal 1320

This theorem is referenced by:  re1luk3  1477