Theorem pm5.61 693

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 394	. . 3 ⊢ (¬ ψ → (φ ↔ (ψ ∨ φ)))
2		orcom 376	. . 3 ⊢ ((ψ ∨ φ) ↔ (φ ∨ ψ))
3	1, 2	syl6rbb 253	. 2 ⊢ (¬ ψ → ((φ ∨ ψ) ↔ φ))
4	3	pm5.32ri 619	1 ⊢ (((φ ∨ ψ) ∧ ¬ ψ) ↔ (φ ∧ ¬ ψ))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  pm5.75  903  nnsucelrlem2  4426