Theorem pm5.61 743

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 698	. . 3 ⊢ (¬ 𝜓 → (𝜑 ↔ (𝜓 ∨ 𝜑)))
2		orcom 682	. . 3 ⊢ ((𝜓 ∨ 𝜑) ↔ (𝜑 ∨ 𝜓))
3	1, 2	syl6rbb 195	. 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
4	3	pm5.32ri 443	1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 102   ↔ wb 103   ∨ wo 664

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm5.75  908  excxor  1314  xrnemnf  9246  xrnepnf  9247