Theorem re1tbw4 1513

Description: tbw-ax4 1468 rederived from merco2 1501.

This theorem, along with re1tbw1 1510, re1tbw2 1511, and re1tbw3 1512, shows that merco2 1501, along with ax-mp 5, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1tbw4	⊢ ( ⊥ → φ)

Proof of Theorem re1tbw4

Step	Hyp	Ref	Expression
1		re1tbw3 1512	. . 3 ⊢ (((φ → φ) → φ) → φ)
2		re1tbw2 1511	. . . 4 ⊢ (φ → ((φ → φ) → φ))
3		re1tbw1 1510	. . . 4 ⊢ ((φ → ((φ → φ) → φ)) → ((((φ → φ) → φ) → φ) → (φ → φ)))
4	2, 3	ax-mp 5	. . 3 ⊢ ((((φ → φ) → φ) → φ) → (φ → φ))
5	1, 4	ax-mp 5	. 2 ⊢ (φ → φ)
6		re1tbw3 1512	. . . . 5 ⊢ (((( ⊥ → φ) → φ) → ( ⊥ → φ)) → ( ⊥ → φ))
7		re1tbw2 1511	. . . . . 6 ⊢ (( ⊥ → φ) → ((( ⊥ → φ) → φ) → ( ⊥ → φ)))
8		re1tbw1 1510	. . . . . 6 ⊢ ((( ⊥ → φ) → ((( ⊥ → φ) → φ) → ( ⊥ → φ))) → ((((( ⊥ → φ) → φ) → ( ⊥ → φ)) → ( ⊥ → φ)) → (( ⊥ → φ) → ( ⊥ → φ))))
9	7, 8	ax-mp 5	. . . . 5 ⊢ ((((( ⊥ → φ) → φ) → ( ⊥ → φ)) → ( ⊥ → φ)) → (( ⊥ → φ) → ( ⊥ → φ)))
10	6, 9	ax-mp 5	. . . 4 ⊢ (( ⊥ → φ) → ( ⊥ → φ))
11		mercolem3 1504	. . . . 5 ⊢ ((( ⊥ → φ) → φ) → (( ⊥ → φ) → ( ⊥ → φ)))
12		merco2 1501	. . . . 5 ⊢ (((( ⊥ → φ) → φ) → (( ⊥ → φ) → ( ⊥ → φ))) → ((( ⊥ → φ) → ( ⊥ → φ)) → ((φ → φ) → ((φ → φ) → ( ⊥ → φ)))))
13	11, 12	ax-mp 5	. . . 4 ⊢ ((( ⊥ → φ) → ( ⊥ → φ)) → ((φ → φ) → ((φ → φ) → ( ⊥ → φ))))
14	10, 13	ax-mp 5	. . 3 ⊢ ((φ → φ) → ((φ → φ) → ( ⊥ → φ)))
15	5, 14	ax-mp 5	. 2 ⊢ ((φ → φ) → ( ⊥ → φ))
16	5, 15	ax-mp 5	1 ⊢ ( ⊥ → 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: (None)