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Mirrors > Home > MPE Home > Th. List > pm5.16 | Structured version Visualization version GIF version |
Description: Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 17-Oct-2013.) |
Ref | Expression |
---|---|
pm5.16 | ⊢ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.18 386 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)) | |
2 | 1 | biimpi 219 | . 2 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) |
3 | imnan 403 | . 2 ⊢ (((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) ↔ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓))) | |
4 | 2, 3 | mpbi 233 | 1 ⊢ ¬ ((𝜑 ↔ 𝜓) ∧ (𝜑 ↔ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 |
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 210 df-an 400 |
This theorem is referenced by: (None) |
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