Theorem con3ALT 1083

Description: Proof of con3 153 from its associated inference con3i 154 that illustrates the use of the weak deduction theorem dedt 1082. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Revised dedt 1082 and elimh 1081. (Revised by Steven Nguyen, 27-Apr-2023.) (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		id 22	. . . 4 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → (if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓))
2	1	notbid 317	. . 3 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → (¬ if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ ¬ 𝜓))
3	2	imbi1d 341	. 2 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((¬ if-((𝜑 → 𝜓), 𝜓, 𝜑) → ¬ 𝜑) ↔ (¬ 𝜓 → ¬ 𝜑)))
4		imbi2 348	. . . 4 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((𝜑 → if-((𝜑 → 𝜓), 𝜓, 𝜑)) ↔ (𝜑 → 𝜓)))
5		imbi2 348	. . . 4 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜑) → ((𝜑 → if-((𝜑 → 𝜓), 𝜓, 𝜑)) ↔ (𝜑 → 𝜑)))
6		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
7	4, 5, 6	elimh 1081	. . 3 ⊢ (𝜑 → if-((𝜑 → 𝜓), 𝜓, 𝜑))
8	7	con3i 154	. 2 ⊢ (¬ if-((𝜑 → 𝜓), 𝜓, 𝜑) → ¬ 𝜑)
9	3, 8	dedt 1082	1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  if-wif 1059

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 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by: (None)