Theorem pm5.61 998

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 888	. . 3 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑))
2		orc 865	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	1, 2	impbid1 224	. 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
4	3	pm5.32ri 574	1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 394   ∨ wo 845

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 206  df-an 395  df-or 846

This theorem is referenced by:  ordtri3  6407  xrnemnf  13132  xrnepnf  13133  hashinfxadd  14380  tltnle  18417  limcdif  25849  ellimc2  25850  limcmpt  25856  limcres  25859  tglineeltr  28507  icorempo  36961  poimirlem14  37238  xrlttri5d  44803