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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 419 | . . 3 ⊢ (¬ 𝜓 → (𝜑 ↔ (𝜓 ∨ 𝜑))) | |
2 | orcom 401 | . . 3 ⊢ ((𝜓 ∨ 𝜑) ↔ (𝜑 ∨ 𝜓)) | |
3 | 1, 2 | syl6rbb 276 | . 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑)) |
4 | 3 | pm5.32ri 668 | 1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 |
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 196 df-or 384 df-an 385 |
This theorem is referenced by: pm5.75OLD 975 ordtri3 5676 xrnemnf 11827 xrnepnf 11828 hashinfxadd 13035 limcdif 23446 ellimc2 23447 limcmpt 23453 limcres 23456 tglineeltr 25326 tltnle 28993 icorempt2 32375 poimirlem14 32593 xrlttri5d 38436 |
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