Theorem tbwlem4 1473

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
tbwlem4	⊢ (((φ → ⊥ ) → ψ) → ((ψ → ⊥ ) → φ))

Proof of Theorem tbwlem4

Step	Hyp	Ref	Expression
1		tbw-ax4 1468	. . . . 5 ⊢ ( ⊥ → ⊥ )
2		tbw-ax1 1465	. . . . . 6 ⊢ ((ψ → ⊥ ) → (( ⊥ → ⊥ ) → (ψ → ⊥ )))
3		tbwlem1 1470	. . . . . 6 ⊢ (((ψ → ⊥ ) → (( ⊥ → ⊥ ) → (ψ → ⊥ ))) → (( ⊥ → ⊥ ) → ((ψ → ⊥ ) → (ψ → ⊥ ))))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (( ⊥ → ⊥ ) → ((ψ → ⊥ ) → (ψ → ⊥ )))
5	1, 4	ax-mp 5	. . . 4 ⊢ ((ψ → ⊥ ) → (ψ → ⊥ ))
6		tbwlem1 1470	. . . 4 ⊢ (((ψ → ⊥ ) → (ψ → ⊥ )) → (ψ → ((ψ → ⊥ ) → ⊥ )))
7	5, 6	ax-mp 5	. . 3 ⊢ (ψ → ((ψ → ⊥ ) → ⊥ ))
8		tbw-ax1 1465	. . . 4 ⊢ (((φ → ⊥ ) → ψ) → ((ψ → ((ψ → ⊥ ) → ⊥ )) → ((φ → ⊥ ) → ((ψ → ⊥ ) → ⊥ ))))
9		tbwlem1 1470	. . . 4 ⊢ ((((φ → ⊥ ) → ψ) → ((ψ → ((ψ → ⊥ ) → ⊥ )) → ((φ → ⊥ ) → ((ψ → ⊥ ) → ⊥ )))) → ((ψ → ((ψ → ⊥ ) → ⊥ )) → (((φ → ⊥ ) → ψ) → ((φ → ⊥ ) → ((ψ → ⊥ ) → ⊥ )))))
10	8, 9	ax-mp 5	. . 3 ⊢ ((ψ → ((ψ → ⊥ ) → ⊥ )) → (((φ → ⊥ ) → ψ) → ((φ → ⊥ ) → ((ψ → ⊥ ) → ⊥ ))))
11	7, 10	ax-mp 5	. 2 ⊢ (((φ → ⊥ ) → ψ) → ((φ → ⊥ ) → ((ψ → ⊥ ) → ⊥ )))
12		tbwlem2 1471	. . 3 ⊢ (((φ → ⊥ ) → ((ψ → ⊥ ) → ⊥ )) → ((((φ → ⊥ ) → φ) → φ) → ((ψ → ⊥ ) → φ)))
13		tbwlem3 1472	. . 3 ⊢ (((((φ → ⊥ ) → φ) → φ) → ((ψ → ⊥ ) → φ)) → ((ψ → ⊥ ) → φ))
14	12, 13	tbwsyl 1469	. 2 ⊢ (((φ → ⊥ ) → ((ψ → ⊥ ) → ⊥ )) → ((ψ → ⊥ ) → φ))
15	11, 14	tbwsyl 1469	1 ⊢ (((φ → ⊥ ) → ψ) → ((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:  tbwlem5  1474  re1luk2  1476