Theorem merco1lem1 1479

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem1	⊢ (φ → ( ⊥ → χ))

Proof of Theorem merco1lem1

Step	Hyp	Ref	Expression
1		merco1 1478	. . . . 5 ⊢ ((((( ⊥ → φ) → (φ → ⊥ )) → (φ → ⊥ )) → ( ⊥ → φ)) → ((( ⊥ → φ) → ⊥ ) → (φ → ⊥ )))
2		merco1 1478	. . . . 5 ⊢ (((((( ⊥ → φ) → (φ → ⊥ )) → (φ → ⊥ )) → ( ⊥ → φ)) → ((( ⊥ → φ) → ⊥ ) → (φ → ⊥ ))) → ((((( ⊥ → φ) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → (φ → ( ⊥ → φ))))
3	1, 2	ax-mp 5	. . . 4 ⊢ ((((( ⊥ → φ) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → (φ → ( ⊥ → φ)))
4		merco1 1478	. . . 4 ⊢ (((((( ⊥ → φ) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → (φ → ( ⊥ → φ))) → (((φ → ( ⊥ → φ)) → ( ⊥ → φ)) → (φ → ( ⊥ → φ))))
5	3, 4	ax-mp 5	. . 3 ⊢ (((φ → ( ⊥ → φ)) → ( ⊥ → φ)) → (φ → ( ⊥ → φ)))
6		merco1 1478	. . . . 5 ⊢ ((((( ⊥ → φ) → (φ → ⊥ )) → ((φ → ( ⊥ → φ)) → ⊥ )) → (φ → ( ⊥ → φ))) → (((φ → ( ⊥ → φ)) → ⊥ ) → (φ → ⊥ )))
7		merco1 1478	. . . . 5 ⊢ (((((( ⊥ → φ) → (φ → ⊥ )) → ((φ → ( ⊥ → φ)) → ⊥ )) → (φ → ( ⊥ → φ))) → (((φ → ( ⊥ → φ)) → ⊥ ) → (φ → ⊥ ))) → (((((φ → ( ⊥ → φ)) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → ((φ → ( ⊥ → φ)) → ( ⊥ → φ))))
8	6, 7	ax-mp 5	. . . 4 ⊢ (((((φ → ( ⊥ → φ)) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → ((φ → ( ⊥ → φ)) → ( ⊥ → φ)))
9		merco1 1478	. . . 4 ⊢ ((((((φ → ( ⊥ → φ)) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → ((φ → ( ⊥ → φ)) → ( ⊥ → φ))) → ((((φ → ( ⊥ → φ)) → ( ⊥ → φ)) → (φ → ( ⊥ → φ))) → (φ → (φ → ( ⊥ → φ)))))
10	8, 9	ax-mp 5	. . 3 ⊢ ((((φ → ( ⊥ → φ)) → ( ⊥ → φ)) → (φ → ( ⊥ → φ))) → (φ → (φ → ( ⊥ → φ))))
11	5, 10	ax-mp 5	. 2 ⊢ (φ → (φ → ( ⊥ → φ)))
12		merco1 1478	. . . . 5 ⊢ ((((( ⊥ → φ) → (φ → ⊥ )) → (φ → ⊥ )) → ( ⊥ → χ)) → ((( ⊥ → χ) → ⊥ ) → (φ → ⊥ )))
13		merco1 1478	. . . . 5 ⊢ (((((( ⊥ → φ) → (φ → ⊥ )) → (φ → ⊥ )) → ( ⊥ → χ)) → ((( ⊥ → χ) → ⊥ ) → (φ → ⊥ ))) → ((((( ⊥ → χ) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → (φ → ( ⊥ → φ))))
14	12, 13	ax-mp 5	. . . 4 ⊢ ((((( ⊥ → χ) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → (φ → ( ⊥ → φ)))
15		merco1 1478	. . . 4 ⊢ (((((( ⊥ → χ) → ⊥ ) → (φ → ⊥ )) → ( ⊥ → φ)) → (φ → ( ⊥ → φ))) → (((φ → ( ⊥ → φ)) → ( ⊥ → χ)) → (φ → ( ⊥ → χ))))
16	14, 15	ax-mp 5	. . 3 ⊢ (((φ → ( ⊥ → φ)) → ( ⊥ → χ)) → (φ → ( ⊥ → χ)))
17		merco1 1478	. . . . 5 ⊢ ((((( ⊥ → χ) → ((φ → (φ → ( ⊥ → φ))) → ⊥ )) → ((φ → ( ⊥ → φ)) → ⊥ )) → (φ → ( ⊥ → χ))) → (((φ → ( ⊥ → χ)) → ⊥ ) → ((φ → (φ → ( ⊥ → φ))) → ⊥ )))
18		merco1 1478	. . . . 5 ⊢ (((((( ⊥ → χ) → ((φ → (φ → ( ⊥ → φ))) → ⊥ )) → ((φ → ( ⊥ → φ)) → ⊥ )) → (φ → ( ⊥ → χ))) → (((φ → ( ⊥ → χ)) → ⊥ ) → ((φ → (φ → ( ⊥ → φ))) → ⊥ ))) → (((((φ → ( ⊥ → χ)) → ⊥ ) → ((φ → (φ → ( ⊥ → φ))) → ⊥ )) → ( ⊥ → χ)) → ((φ → ( ⊥ → φ)) → ( ⊥ → χ))))
19	17, 18	ax-mp 5	. . . 4 ⊢ (((((φ → ( ⊥ → χ)) → ⊥ ) → ((φ → (φ → ( ⊥ → φ))) → ⊥ )) → ( ⊥ → χ)) → ((φ → ( ⊥ → φ)) → ( ⊥ → χ)))
20		merco1 1478	. . . 4 ⊢ ((((((φ → ( ⊥ → χ)) → ⊥ ) → ((φ → (φ → ( ⊥ → φ))) → ⊥ )) → ( ⊥ → χ)) → ((φ → ( ⊥ → φ)) → ( ⊥ → χ))) → ((((φ → ( ⊥ → φ)) → ( ⊥ → χ)) → (φ → ( ⊥ → χ))) → ((φ → (φ → ( ⊥ → φ))) → (φ → ( ⊥ → χ)))))
21	19, 20	ax-mp 5	. . 3 ⊢ ((((φ → ( ⊥ → φ)) → ( ⊥ → χ)) → (φ → ( ⊥ → χ))) → ((φ → (φ → ( ⊥ → φ))) → (φ → ( ⊥ → χ))))
22	16, 21	ax-mp 5	. 2 ⊢ ((φ → (φ → ( ⊥ → φ))) → (φ → ( ⊥ → χ)))
23	11, 22	ax-mp 5	1 ⊢ (φ → ( ⊥ → χ))

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:  retbwax4  1480  retbwax2  1481