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Mirrors > Home > MPE Home > Th. List > nannan | Structured version Visualization version GIF version |
Description: Nested alternative denials. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 26-Jun-2020.) |
Ref | Expression |
---|---|
nannan | ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnan2 1490 | . 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒))) | |
2 | nanan 1489 | . . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ ¬ (𝜓 ⊼ 𝜒)) | |
3 | 2 | imbi2i 335 | . 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ⊼ 𝜒))) |
4 | 1, 3 | bitr4i 277 | 1 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ⊼ wnan 1487 |
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 206 df-an 395 df-nan 1488 |
This theorem is referenced by: nanim 1494 nanbi 1496 nanass 1506 nic-mp 1671 nic-ax 1673 waj-ax 35602 lukshef-ax2 35603 arg-ax 35604 |
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