Theorem re1luk3 1477

Description: luk-3 1422 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1475 and re1luk2 1476 proves that tbw-ax1 1465, tbw-ax2 1466, tbw-ax3 1467, and tbw-ax4 1468, with ax-mp 5 can be used as a complete axiom system for all of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1luk3	⊢ (φ → (¬ φ → ψ))

Proof of Theorem re1luk3

Step	Hyp	Ref	Expression
1		tbw-negdf 1464	. . 3 ⊢ (((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ )
2		tbwlem5 1474	. . 3 ⊢ ((((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ ) → (¬ φ → (φ → ⊥ )))
3	1, 2	ax-mp 5	. 2 ⊢ (¬ φ → (φ → ⊥ ))
4		tbw-ax4 1468	. . . 4 ⊢ ( ⊥ → ψ)
5		tbw-ax1 1465	. . . . 5 ⊢ ((φ → ⊥ ) → (( ⊥ → ψ) → (φ → ψ)))
6		tbwlem1 1470	. . . . 5 ⊢ (((φ → ⊥ ) → (( ⊥ → ψ) → (φ → ψ))) → (( ⊥ → ψ) → ((φ → ⊥ ) → (φ → ψ))))
7	5, 6	ax-mp 5	. . . 4 ⊢ (( ⊥ → ψ) → ((φ → ⊥ ) → (φ → ψ)))
8	4, 7	ax-mp 5	. . 3 ⊢ ((φ → ⊥ ) → (φ → ψ))
9		tbwlem1 1470	. . 3 ⊢ (((φ → ⊥ ) → (φ → ψ)) → (φ → ((φ → ⊥ ) → ψ)))
10	8, 9	ax-mp 5	. 2 ⊢ (φ → ((φ → ⊥ ) → ψ))
11		tbw-ax1 1465	. 2 ⊢ ((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ψ) → (¬ φ → ψ)))
12	3, 10, 11	mpsyl 59	1 ⊢ (φ → (¬ φ → ψ))

This proof depends on syntax axioms:  ¬ wn 3   → 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: (None)