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Theorem con3ALT 1082
 Description: Proof of con3 156 from its associated inference con3i 157 that illustrates the use of the weak deduction theorem dedt 1081. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Revised dedt 1081 and elimh 1080. (Revised by Steven Nguyen, 27-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
con3ALT ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3ALT
StepHypRef Expression
1 id 22 . . . 4 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → (if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓))
21notbid 321 . . 3 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → (¬ if-((𝜑𝜓), 𝜓, 𝜑) ↔ ¬ 𝜓))
32imbi1d 345 . 2 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((¬ if-((𝜑𝜓), 𝜓, 𝜑) → ¬ 𝜑) ↔ (¬ 𝜓 → ¬ 𝜑)))
4 imbi2 352 . . . 4 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((𝜑 → if-((𝜑𝜓), 𝜓, 𝜑)) ↔ (𝜑𝜓)))
5 imbi2 352 . . . 4 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜑) → ((𝜑 → if-((𝜑𝜓), 𝜓, 𝜑)) ↔ (𝜑𝜑)))
6 id 22 . . . 4 (𝜑𝜑)
74, 5, 6elimh 1080 . . 3 (𝜑 → if-((𝜑𝜓), 𝜓, 𝜑))
87con3i 157 . 2 (¬ if-((𝜑𝜓), 𝜓, 𝜑) → ¬ 𝜑)
93, 8dedt 1081 1 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  if-wif 1058 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 210  df-an 400  df-or 845  df-ifp 1059 This theorem is referenced by: (None)
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