Theorem pm5.61 806

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 756	. . 3 ⊢ (¬ 𝜓 → (𝜑 ↔ (𝜓 ∨ 𝜑)))
2		orcom 740	. . 3 ⊢ ((𝜓 ∨ 𝜑) ↔ (𝜑 ∨ 𝜓))
3	1, 2	bitr2di 197	. 2 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) ↔ 𝜑))
4	3	pm5.32ri 459	1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  pm5.75  975  excxor  1427  xrnemnf  10179  xrnepnf  10180