Theorem pm5.54 870

Description: Theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 7-Nov-2013.)

Assertion

Ref	Expression
pm5.54	⊢ (((φ ∧ ψ) ↔ φ) ∨ ((φ ∧ ψ) ↔ ψ))

Proof of Theorem pm5.54

Step	Hyp	Ref	Expression
1		iba 489	. . . . 5 ⊢ (ψ → (φ ↔ (φ ∧ ψ)))
2	1	bicomd 192	. . . 4 ⊢ (ψ → ((φ ∧ ψ) ↔ φ))
3	2	adantl 452	. . 3 ⊢ ((φ ∧ ψ) → ((φ ∧ ψ) ↔ φ))
4	3, 2	pm5.21ni 341	. 2 ⊢ (¬ ((φ ∧ ψ) ↔ φ) → ((φ ∧ ψ) ↔ ψ))
5	4	orri 365	1 ⊢ (((φ ∧ ψ) ↔ φ) ∨ ((φ ∧ ψ) ↔ ψ))

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

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)