Theorem pm5.61 1014

Description: Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)

Assertion

Ref	Expression
pm5.61	⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Proof of Theorem pm5.61

Step	Hyp	Ref	Expression
1		orel2 901	. . 3 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑))
2		orc 878	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	1, 2	impbid1 227	. 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
4	3	pm5.32ri 583	1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399   ∨ wo 858

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 209  df-an 400  df-or 859

This theorem is referenced by:  ordtri3  6382  xrnemnf  13119  xrnepnf  13120  hashinfxadd  14398  tltnle  18452  limcdif  25938  ellimc2  25939  limcmpt  25945  limcres  25948  tglineeltr  28800  icorempo  37845  poimirlem14  38133  xrlttri5d  45863