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Mirrors > Home > MPE Home > Th. List > nanbi | Structured version Visualization version GIF version |
Description: Biconditional in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.) |
Ref | Expression |
---|---|
nanbi | ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi3 1047 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) | |
2 | df-or 845 | . . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓))) | |
3 | df-nan 1487 | . . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
4 | 3 | bicomi 223 | . . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (𝜑 ⊼ 𝜓)) |
5 | nannot 1494 | . . . . 5 ⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑)) | |
6 | nannot 1494 | . . . . 5 ⊢ (¬ 𝜓 ↔ (𝜓 ⊼ 𝜓)) | |
7 | 5, 6 | anbi12i 627 | . . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓))) |
8 | 4, 7 | imbi12i 351 | . . 3 ⊢ ((¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ⊼ 𝜓) → ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓)))) |
9 | 1, 2, 8 | 3bitri 297 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ⊼ 𝜓) → ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓)))) |
10 | nannan 1492 | . 2 ⊢ (((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) ↔ ((𝜑 ⊼ 𝜓) → ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓)))) | |
11 | 9, 10 | bitr4i 277 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ⊼ wnan 1486 |
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 397 df-or 845 df-nan 1487 |
This theorem is referenced by: nic-dfim 1676 nic-dfneg 1677 |
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