Theorem con3ALT 1026

Description: Proof of con3 148 from its associated inference con3i 149 that illustrates the use of the weak deduction theorem dedt 1025. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
con3ALT	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3ALT

Step	Hyp	Ref	Expression
1		bicom1 210	. . . 4 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → (𝜓 ↔ if-((𝜑 → 𝜓), 𝜓, 𝜑)))
2	1	notbid 307	. . 3 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → (¬ 𝜓 ↔ ¬ if-((𝜑 → 𝜓), 𝜓, 𝜑)))
3	2	imbi1d 330	. 2 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((¬ 𝜓 → ¬ 𝜑) ↔ (¬ if-((𝜑 → 𝜓), 𝜓, 𝜑) → ¬ 𝜑)))
4	1	imbi2d 329	. . . 4 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((𝜑 → 𝜓) ↔ (𝜑 → if-((𝜑 → 𝜓), 𝜓, 𝜑))))
5		bicom1 210	. . . . 5 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜑) → (𝜑 ↔ if-((𝜑 → 𝜓), 𝜓, 𝜑)))
6	5	imbi2d 329	. . . 4 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜑) → ((𝜑 → 𝜑) ↔ (𝜑 → if-((𝜑 → 𝜓), 𝜓, 𝜑))))
7		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
8	4, 6, 7	elimh 1024	. . 3 ⊢ (𝜑 → if-((𝜑 → 𝜓), 𝜓, 𝜑))
9	8	con3i 149	. 2 ⊢ (¬ if-((𝜑 → 𝜓), 𝜓, 𝜑) → ¬ 𝜑)
10	3, 9	dedt 1025	1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  if-wif 1006

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 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by: (None)