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Mirrors > Home > MPE Home > Th. List > con3ALT | Structured version Visualization version GIF version |
Description: Proof of con3 149 from its associated inference con3i 150 that illustrates the use of the weak deduction theorem dedt 1031. (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.) |
Ref | Expression |
---|---|
con3ALT | ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bicom1 211 | . . . 4 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → (𝜓 ↔ if-((𝜑 → 𝜓), 𝜓, 𝜑))) | |
2 | 1 | notbid 308 | . . 3 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → (¬ 𝜓 ↔ ¬ if-((𝜑 → 𝜓), 𝜓, 𝜑))) |
3 | 2 | imbi1d 331 | . 2 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((¬ 𝜓 → ¬ 𝜑) ↔ (¬ if-((𝜑 → 𝜓), 𝜓, 𝜑) → ¬ 𝜑))) |
4 | 1 | imbi2d 330 | . . . 4 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜓) → ((𝜑 → 𝜓) ↔ (𝜑 → if-((𝜑 → 𝜓), 𝜓, 𝜑)))) |
5 | bicom1 211 | . . . . 5 ⊢ ((if-((𝜑 → 𝜓), 𝜓, 𝜑) ↔ 𝜑) → (𝜑 ↔ if-((𝜑 → 𝜓), 𝜓, 𝜑))) | |
6 | 5 | imbi2d 330 | . . . 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 1030 | . . 3 ⊢ (𝜑 → if-((𝜑 → 𝜓), 𝜓, 𝜑)) |
9 | 8 | con3i 150 | . 2 ⊢ (¬ if-((𝜑 → 𝜓), 𝜓, 𝜑) → ¬ 𝜑) |
10 | 3, 9 | dedt 1031 | 1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 if-wif 1012 |
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 197 df-or 385 df-an 386 df-ifp 1013 |
This theorem is referenced by: (None) |
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