Theorem pm5.61 749

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		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))

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