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Mirrors > Home > MPE Home > Th. List > Mathboxes > imnand2 | Structured version Visualization version GIF version |
Description: An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.) |
Ref | Expression |
---|---|
imnand2 | ⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nannot 1453 | . . . 4 ⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑)) | |
2 | nannot 1453 | . . . 4 ⊢ (¬ 𝜓 ↔ (𝜓 ⊼ 𝜓)) | |
3 | 1, 2 | anbi12i 733 | . . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓))) |
4 | 3 | notbii 310 | . 2 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓))) |
5 | iman 440 | . 2 ⊢ ((¬ 𝜑 → 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)) | |
6 | df-nan 1448 | . 2 ⊢ (((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)) ↔ ¬ ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓))) | |
7 | 4, 5, 6 | 3bitr4i 292 | 1 ⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ⊼ wnan 1447 |
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-an 386 df-nan 1448 |
This theorem is referenced by: (None) |
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