Theorem pm5.61 1002

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 890	. . 3 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑))
2		orc 867	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	1, 2	impbid1 225	. 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
4	3	pm5.32ri 575	1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847

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 207  df-an 396  df-or 848

This theorem is referenced by:  ordtri3  6353  xrnemnf  13031  xrnepnf  13032  hashinfxadd  14308  tltnle  18343  limcdif  25833  ellimc2  25834  limcmpt  25840  limcres  25843  tglineeltr  28703  icorempo  37556  poimirlem14  37835  xrlttri5d  45542