Theorem pm4.42 926

Description: Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Roy F. Longton, 21-Jun-2005.)

Assertion

Ref	Expression
pm4.42	⊢ (φ ↔ ((φ ∧ ψ) ∨ (φ ∧ ¬ ψ)))

Proof of Theorem pm4.42

Step	Hyp	Ref	Expression
1		dedlema 920	. 2 ⊢ (ψ → (φ ↔ ((φ ∧ ψ) ∨ (φ ∧ ¬ ψ))))
2		dedlemb 921	. 2 ⊢ (¬ ψ → (φ ↔ ((φ ∧ ψ) ∨ (φ ∧ ¬ ψ))))
3	1, 2	pm2.61i 156	1 ⊢ (φ ↔ ((φ ∧ ψ) ∨ (φ ∧ ¬ ψ)))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8

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

This theorem is referenced by:  inundif  3628