Theorem merco2 1501

Description: A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1478. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco2	⊢ (((φ → ψ) → (( ⊥ → χ) → θ)) → ((θ → φ) → (τ → (η → φ))))

Proof of Theorem merco2

Step	Hyp	Ref	Expression
1		falim 1328	. . . . . 6 ⊢ ( ⊥ → χ)
2		pm2.04 76	. . . . . 6 ⊢ (((φ → ψ) → (( ⊥ → χ) → θ)) → (( ⊥ → χ) → ((φ → ψ) → θ)))
3	1, 2	mpi 16	. . . . 5 ⊢ (((φ → ψ) → (( ⊥ → χ) → θ)) → ((φ → ψ) → θ))
4		jarl 155	. . . . . 6 ⊢ (((φ → ψ) → θ) → (¬ φ → θ))
5		idd 21	. . . . . 6 ⊢ (((φ → ψ) → θ) → (θ → θ))
6	4, 5	jad 154	. . . . 5 ⊢ (((φ → ψ) → θ) → ((φ → θ) → θ))
7		looinv 174	. . . . 5 ⊢ (((φ → θ) → θ) → ((θ → φ) → φ))
8	3, 6, 7	3syl 18	. . . 4 ⊢ (((φ → ψ) → (( ⊥ → χ) → θ)) → ((θ → φ) → φ))
9	8	a1dd 42	. . 3 ⊢ (((φ → ψ) → (( ⊥ → χ) → θ)) → ((θ → φ) → (τ → φ)))
10	9	a1i 10	. 2 ⊢ (η → (((φ → ψ) → (( ⊥ → χ) → θ)) → ((θ → φ) → (τ → φ))))
11	10	com4l 78	1 ⊢ (((φ → ψ) → (( ⊥ → χ) → θ)) → ((θ → φ) → (τ → (η → 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:  mercolem1  1502  mercolem2  1503  mercolem3  1504  mercolem4  1505  mercolem5  1506  mercolem6  1507  mercolem7  1508  mercolem8  1509  re1tbw4  1513