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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 890 | . . 3 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑)) | |
2 | orc 866 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 1, 2 | impbid1 224 | . 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑)) |
4 | 3 | pm5.32ri 577 | 1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 |
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 398 df-or 847 |
This theorem is referenced by: ordtri3 6358 xrnemnf 13045 xrnepnf 13046 hashinfxadd 14292 tltnle 18318 limcdif 25256 ellimc2 25257 limcmpt 25263 limcres 25266 tglineeltr 27615 icorempo 35851 poimirlem14 36121 xrlttri5d 43591 |
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