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| Mirrors > Home > MPE Home > Th. List > pm5.61 | Structured version Visualization version GIF version | ||
| Description: Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.) |
| Ref | Expression |
|---|---|
| pm5.61 | ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | orel2 891 | . . 3 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑)) | |
| 2 | orc 868 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
| 3 | 1, 2 | impbid1 225 | . 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑)) |
| 4 | 3 | pm5.32ri 575 | 1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 848 |
| 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 207 df-an 396 df-or 849 |
| This theorem is referenced by: ordtri3 6420 xrnemnf 13159 xrnepnf 13160 hashinfxadd 14424 tltnle 18467 limcdif 25911 ellimc2 25912 limcmpt 25918 limcres 25921 tglineeltr 28639 icorempo 37352 poimirlem14 37641 xrlttri5d 45295 |
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