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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-nannan | Structured version Visualization version GIF version |
Description: Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.) |
Ref | Expression |
---|---|
wl-nannan | ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-dfnan2 33296 | . 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒))) | |
2 | nanan 1449 | . . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 ⊼ 𝜒)) | |
3 | 2 | imbi2i 326 | . 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒))) |
4 | 1, 3 | bitr4i 267 | 1 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒))) |
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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