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Mirrors > Home > MPE Home > Th. List > Mathboxes > bicontr | Structured version Visualization version GIF version |
Description: Biimplication of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
Ref | Expression |
---|---|
bicontr | ⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biid 250 | . . 3 ⊢ (𝜑 ↔ 𝜑) | |
2 | notbinot1 33048 | . . 3 ⊢ (¬ (¬ 𝜑 ↔ 𝜑) ↔ (𝜑 ↔ 𝜑)) | |
3 | 1, 2 | mpbir 220 | . 2 ⊢ ¬ (¬ 𝜑 ↔ 𝜑) |
4 | 3 | bifal 1488 | 1 ⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ⊥wfal 1480 |
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 196 df-tru 1478 df-fal 1481 |
This theorem is referenced by: (None) |
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