Theorem merco1lem16 1497

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem16	⊢ (((φ → (ψ → χ)) → τ) → ((φ → χ) → τ))

Proof of Theorem merco1lem16

Step	Hyp	Ref	Expression
1		merco1lem15 1496	. . 3 ⊢ ((φ → χ) → (φ → (ψ → χ)))
2		merco1lem11 1492	. . 3 ⊢ (((φ → χ) → (φ → (ψ → χ))) → ((((τ → φ) → ((φ → χ) → ⊥ )) → ⊥ ) → (φ → (ψ → χ))))
3	1, 2	ax-mp 5	. 2 ⊢ ((((τ → φ) → ((φ → χ) → ⊥ )) → ⊥ ) → (φ → (ψ → χ)))
4		merco1 1478	. 2 ⊢ (((((τ → φ) → ((φ → χ) → ⊥ )) → ⊥ ) → (φ → (ψ → χ))) → (((φ → (ψ → χ)) → τ) → ((φ → χ) → τ)))
5	3, 4	ax-mp 5	1 ⊢ (((φ → (ψ → χ)) → τ) → ((φ → χ) → τ))

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:  merco1lem17  1498  retbwax1  1500