Theorem dedlem0b 1045

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		pm2.21 123	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → (𝜒 ∧ 𝜑)))
2	1	imim2d 57	. . 3 ⊢ (¬ 𝜑 → ((𝜓 → 𝜑) → (𝜓 → (𝜒 ∧ 𝜑))))
3	2	com23 86	. 2 ⊢ (¬ 𝜑 → (𝜓 → ((𝜓 → 𝜑) → (𝜒 ∧ 𝜑))))
4		pm2.21 123	. . . . 5 ⊢ (¬ 𝜓 → (𝜓 → 𝜑))
5		simpr 484	. . . . 5 ⊢ ((𝜒 ∧ 𝜑) → 𝜑)
6	4, 5	imim12i 62	. . . 4 ⊢ (((𝜓 → 𝜑) → (𝜒 ∧ 𝜑)) → (¬ 𝜓 → 𝜑))
7	6	con1d 145	. . 3 ⊢ (((𝜓 → 𝜑) → (𝜒 ∧ 𝜑)) → (¬ 𝜑 → 𝜓))
8	7	com12 32	. 2 ⊢ (¬ 𝜑 → (((𝜓 → 𝜑) → (𝜒 ∧ 𝜑)) → 𝜓))
9	3, 8	impbid 212	1 ⊢ (¬ 𝜑 → (𝜓 ↔ ((𝜓 → 𝜑) → (𝜒 ∧ 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by: (None)