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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 | biorf 420 | . . 3 ⊢ (¬ 𝜓 → (𝜑 ↔ (𝜓 ∨ 𝜑))) | |
2 | orcom 402 | . . 3 ⊢ ((𝜓 ∨ 𝜑) ↔ (𝜑 ∨ 𝜓)) | |
3 | 1, 2 | syl6rbb 277 | . 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑)) |
4 | 3 | pm5.32ri 670 | 1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∨ wo 383 ∧ wa 384 |
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 197 df-or 385 df-an 386 |
This theorem is referenced by: pm5.75OLD 979 ordtri3 5759 xrnemnf 11951 xrnepnf 11952 hashinfxadd 13174 limcdif 23640 ellimc2 23641 limcmpt 23647 limcres 23650 tglineeltr 25526 tltnle 29662 icorempt2 33199 poimirlem14 33423 xrlttri5d 39495 |
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