Theorem dedlem0b 568

Description: Lemma for an alternate version of weak deduction theorem.

Assertion

Ref	Expression
dedlem0b	⊢ (¬ φ → (ψ ↔ ((ψ → φ) → (χ ∧ φ))))

Proof of Theorem dedlem0b

Step	Hyp	Ref	Expression
1		pm2.21 71	. . . 4 ⊢ (¬ φ → (φ → (χ ∧ φ)))
2	1	syl3d 26	. . 3 ⊢ (¬ φ → ((ψ → φ) → (ψ → (χ ∧ φ))))
3	2	com23 32	. 2 ⊢ (¬ φ → (ψ → ((ψ → φ) → (χ ∧ φ))))
4		pm2.21 71	. . . . 5 ⊢ (¬ ψ → (ψ → φ))
5		pm3.27 260	. . . . 5 ⊢ ((χ ∧ φ) → φ)
6	4, 5	syl34 20	. . . 4 ⊢ (((ψ → φ) → (χ ∧ φ)) → (¬ ψ → φ))
7	6	con1d 85	. . 3 ⊢ (((ψ → φ) → (χ ∧ φ)) → (¬ φ → ψ))
8	7	com12 13	. 2 ⊢ (¬ φ → (((ψ → φ) → (χ ∧ φ)) → ψ))
9	3, 8	impbid 397	1 ⊢ (¬ φ → (ψ ↔ ((ψ → φ) → (χ ∧ φ))))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org