Theorem merco1lem6 1486

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
merco1lem6	⊢ ((φ → (φ → ψ)) → (χ → (φ → ψ)))

Proof of Theorem merco1lem6

Step	Hyp	Ref	Expression
1		merco1lem5 1485	. . . . 5 ⊢ (((((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → ((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ))
2		merco1lem3 1483	. . . . 5 ⊢ ((((((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → ((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ )) → ((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ )))
3	1, 2	ax-mp 5	. . . 4 ⊢ ((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ ))
4		merco1lem5 1485	. . . 4 ⊢ (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ )) → ((φ → ψ) → (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ )))
5	3, 4	ax-mp 5	. . 3 ⊢ ((φ → ψ) → (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ ))
6		merco1lem3 1483	. . 3 ⊢ (((φ → ψ) → (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → ⊥ )) → (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → φ))
7	5, 6	ax-mp 5	. 2 ⊢ (((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → φ)
8		merco1 1478	. 2 ⊢ ((((((φ → ψ) → ⊥ ) → (χ → ⊥ )) → ⊥ ) → φ) → ((φ → (φ → ψ)) → (χ → (φ → ψ))))
9	7, 8	ax-mp 5	1 ⊢ ((φ → (φ → ψ)) → (χ → (φ → ψ)))

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:  merco1lem7  1487  merco1lem8  1489