Theorem mercolem3 1504

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mercolem3	⊢ ((ψ → χ) → (ψ → (φ → χ)))

Proof of Theorem mercolem3

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 ⊢ (((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ))))
2		merco2 1501	. . . 4 ⊢ (((χ → φ) → (( ⊥ → φ) → ψ)) → ((ψ → χ) → (ψ → (φ → χ))))
3		mercolem2 1503	. . . . . . 7 ⊢ (((ψ → (φ → χ)) → ψ) → (( ⊥ → φ) → (( ⊥ → φ) → ψ)))
4		merco2 1501	. . . . . . 7 ⊢ ((((ψ → (φ → χ)) → ψ) → (( ⊥ → φ) → (( ⊥ → φ) → ψ))) → (((( ⊥ → φ) → ψ) → (ψ → (φ → χ))) → (( ⊥ → φ) → ((ψ → χ) → (ψ → (φ → χ))))))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ (((( ⊥ → φ) → ψ) → (ψ → (φ → χ))) → (( ⊥ → φ) → ((ψ → χ) → (ψ → (φ → χ)))))
6		merco2 1501	. . . . . 6 ⊢ ((((( ⊥ → φ) → ψ) → (ψ → (φ → χ))) → (( ⊥ → φ) → ((ψ → χ) → (ψ → (φ → χ))))) → ((((ψ → χ) → (ψ → (φ → χ))) → (( ⊥ → φ) → ψ)) → (( ⊥ → φ) → ((χ → φ) → (( ⊥ → φ) → ψ)))))
7	5, 6	ax-mp 5	. . . . 5 ⊢ ((((ψ → χ) → (ψ → (φ → χ))) → (( ⊥ → φ) → ψ)) → (( ⊥ → φ) → ((χ → φ) → (( ⊥ → φ) → ψ))))
8		merco2 1501	. . . . 5 ⊢ (((((ψ → χ) → (ψ → (φ → χ))) → (( ⊥ → φ) → ψ)) → (( ⊥ → φ) → ((χ → φ) → (( ⊥ → φ) → ψ)))) → ((((χ → φ) → (( ⊥ → φ) → ψ)) → ((ψ → χ) → (ψ → (φ → χ)))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((ψ → χ) → (ψ → (φ → χ)))))))
9	7, 8	ax-mp 5	. . . 4 ⊢ ((((χ → φ) → (( ⊥ → φ) → ψ)) → ((ψ → χ) → (ψ → (φ → χ)))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((ψ → χ) → (ψ → (φ → χ))))))
10	2, 9	ax-mp 5	. . 3 ⊢ ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((ψ → χ) → (ψ → (φ → χ)))))
11	1, 10	ax-mp 5	. 2 ⊢ ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((ψ → χ) → (ψ → (φ → χ))))
12	1, 11	ax-mp 5	1 ⊢ ((ψ → χ) → (ψ → (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:  mercolem4  1505  mercolem7  1508  mercolem8  1509  re1tbw1  1510  re1tbw4  1513