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Theorem con3ALT 1076
Description: Proof of con3 156 from its associated inference con3i 157 that illustrates the use of the weak deduction theorem dedt 1074. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Revised dedt 1074 and elimh 1072. (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 319 . . 3 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → (¬ if-((𝜑𝜓), 𝜓, 𝜑) ↔ ¬ 𝜓))
32imbi1d 343 . 2 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((¬ if-((𝜑𝜓), 𝜓, 𝜑) → ¬ 𝜑) ↔ (¬ 𝜓 → ¬ 𝜑)))
4 imbi2 350 . . . 4 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((𝜑 → if-((𝜑𝜓), 𝜓, 𝜑)) ↔ (𝜑𝜓)))
5 imbi2 350 . . . 4 ((if-((𝜑𝜓), 𝜓, 𝜑) ↔ 𝜑) → ((𝜑 → if-((𝜑𝜓), 𝜓, 𝜑)) ↔ (𝜑𝜑)))
6 id 22 . . . 4 (𝜑𝜑)
74, 5, 6elimh 1072 . . 3 (𝜑 → if-((𝜑𝜓), 𝜓, 𝜑))
87con3i 157 . 2 (¬ if-((𝜑𝜓), 𝜓, 𝜑) → ¬ 𝜑)
93, 8dedt 1074 1 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  if-wif 1054
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 208  df-an 397  df-or 842  df-ifp 1055
This theorem is referenced by: (None)
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