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Theorem con3ALT 1025
Description: Proof of con3 147 from its associated inference con3i 148 that illustrates the use of the weak deduction theorem dedt 1024. (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 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3ALT
StepHypRef Expression
1 bicom1 209 . . . 4 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → (𝜓 ↔ if-((𝜑𝜓), 𝜓, 𝜑)))
21notbid 306 . . 3 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → (¬ 𝜓 ↔ ¬ if-((𝜑𝜓), 𝜓, 𝜑)))
32imbi1d 329 . 2 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((¬ 𝜓 → ¬ 𝜑) ↔ (¬ if-((𝜑𝜓), 𝜓, 𝜑) → ¬ 𝜑)))
41imbi2d 328 . . . 4 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((𝜑𝜓) ↔ (𝜑 → if-((𝜑𝜓), 𝜓, 𝜑))))
5 bicom1 209 . . . . 5 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜑) → (𝜑 ↔ if-((𝜑𝜓), 𝜓, 𝜑)))
65imbi2d 328 . . . 4 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜑) → ((𝜑𝜑) ↔ (𝜑 → if-((𝜑𝜓), 𝜓, 𝜑))))
7 id 22 . . . 4 (𝜑𝜑)
84, 6, 7elimh 1023 . . 3 (𝜑 → if-((𝜑𝜓), 𝜓, 𝜑))
98con3i 148 . 2 (¬ if-((𝜑𝜓), 𝜓, 𝜑) → ¬ 𝜑)
103, 9dedt 1024 1 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  if-wif 1005
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 195  df-or 383  df-an 384  df-ifp 1006
This theorem is referenced by: (None)
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