Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 884 | . . 3 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑)) | |
2 | orc 861 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 1, 2 | impbid1 226 | . 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑)) |
4 | 3 | pm5.32ri 576 | 1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∨ wo 841 |
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 208 df-an 397 df-or 842 |
This theorem is referenced by: ordtri3 6220 xrnemnf 12500 xrnepnf 12501 hashinfxadd 13734 limcdif 24401 ellimc2 24402 limcmpt 24408 limcres 24411 tglineeltr 26344 tltnle 30576 icorempo 34514 poimirlem14 34787 xrlttri5d 41425 |
Copyright terms: Public domain | W3C validator |