Theorem dedlem0b 919

Description: Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)

Assertion

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

Proof of Theorem dedlem0b

Step	Hyp	Ref	Expression
1		pm2.21 100	. . . 4 ⊢ (¬ φ → (φ → (χ ∧ φ)))
2	1	imim2d 48	. . 3 ⊢ (¬ φ → ((ψ → φ) → (ψ → (χ ∧ φ))))
3	2	com23 72	. 2 ⊢ (¬ φ → (ψ → ((ψ → φ) → (χ ∧ φ))))
4		pm2.21 100	. . . . 5 ⊢ (¬ ψ → (ψ → φ))
5		simpr 447	. . . . 5 ⊢ ((χ ∧ φ) → φ)
6	4, 5	imim12i 53	. . . 4 ⊢ (((ψ → φ) → (χ ∧ φ)) → (¬ ψ → φ))
7	6	con1d 116	. . 3 ⊢ (((ψ → φ) → (χ ∧ φ)) → (¬ φ → ψ))
8	7	com12 27	. 2 ⊢ (¬ φ → (((ψ → φ) → (χ ∧ φ)) → ψ))
9	3, 8	impbid 183	1 ⊢ (¬ φ → (ψ ↔ ((ψ → φ) → (χ ∧ φ))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358

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-an 360

This theorem is referenced by: (None)