Theorem con3OLD 1035

Description: Old version of con3ALT 1032. Obsolete as of 16-Mar-2021. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
con3OLD	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3OLD

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ ((𝜓 ↔ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))) → (𝜓 ↔ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))))
2	1	notbid 308	. . 3 ⊢ ((𝜓 ↔ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))) → (¬ 𝜓 ↔ ¬ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))))
3	2	imbi1d 331	. 2 ⊢ ((𝜓 ↔ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))) → ((¬ 𝜓 → ¬ 𝜑) ↔ (¬ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓))) → ¬ 𝜑)))
4	1	imbi2d 330	. . . 4 ⊢ ((𝜓 ↔ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))) → ((𝜑 → 𝜓) ↔ (𝜑 → ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓))))))
5		id 22	. . . . 5 ⊢ ((𝜑 ↔ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))) → (𝜑 ↔ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))))
6	5	imbi2d 330	. . . 4 ⊢ ((𝜑 ↔ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓)))) → ((𝜑 → 𝜑) ↔ (𝜑 → ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓))))))
7		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
8	4, 6, 7	elimhOLD 1033	. . 3 ⊢ (𝜑 → ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓))))
9	8	con3i 150	. 2 ⊢ (¬ ((𝜓 ∧ (𝜑 → 𝜓)) ∨ (𝜑 ∧ ¬ (𝜑 → 𝜓))) → ¬ 𝜑)
10	3, 9	dedtOLD 1034	1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by: (None)