Theorem tbwlem2 1471

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
tbwlem2	⊢ ((φ → (ψ → ⊥ )) → (((φ → χ) → θ) → (ψ → θ)))

Proof of Theorem tbwlem2

Step	Hyp	Ref	Expression
1		tbw-ax4 1468	. . . . 5 ⊢ ( ⊥ → χ)
2		tbw-ax1 1465	. . . . . 6 ⊢ ((ψ → ⊥ ) → (( ⊥ → χ) → (ψ → χ)))
3		tbwlem1 1470	. . . . . 6 ⊢ (((ψ → ⊥ ) → (( ⊥ → χ) → (ψ → χ))) → (( ⊥ → χ) → ((ψ → ⊥ ) → (ψ → χ))))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (( ⊥ → χ) → ((ψ → ⊥ ) → (ψ → χ)))
5	1, 4	ax-mp 5	. . . 4 ⊢ ((ψ → ⊥ ) → (ψ → χ))
6		tbwlem1 1470	. . . 4 ⊢ (((ψ → ⊥ ) → (ψ → χ)) → (ψ → ((ψ → ⊥ ) → χ)))
7	5, 6	ax-mp 5	. . 3 ⊢ (ψ → ((ψ → ⊥ ) → χ))
8		tbw-ax1 1465	. . 3 ⊢ ((φ → (ψ → ⊥ )) → (((ψ → ⊥ ) → χ) → (φ → χ)))
9		tbw-ax1 1465	. . 3 ⊢ ((ψ → ((ψ → ⊥ ) → χ)) → ((((ψ → ⊥ ) → χ) → (φ → χ)) → (ψ → (φ → χ))))
10	7, 8, 9	mpsyl 59	. 2 ⊢ ((φ → (ψ → ⊥ )) → (ψ → (φ → χ)))
11		tbw-ax1 1465	. 2 ⊢ ((ψ → (φ → χ)) → (((φ → χ) → θ) → (ψ → θ)))
12	10, 11	tbwsyl 1469	1 ⊢ ((φ → (ψ → ⊥ )) → (((φ → χ) → θ) → (ψ → θ)))

This proof depends on syntax axioms:   → wi 4   ⊥ wfal 1317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320

This theorem is used by:  tbwlem4  1473