Theorem dfbi3 863

Description: An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)

Assertion

Ref	Expression
dfbi3	⊢ ((φ ↔ ψ) ↔ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)))

Proof of Theorem dfbi3

Step	Hyp	Ref	Expression
1		xor 861	. 2 ⊢ (¬ (φ ↔ ¬ ψ) ↔ ((φ ∧ ¬ ¬ ψ) ∨ (¬ ψ ∧ ¬ φ)))
2		pm5.18 345	. 2 ⊢ ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ))
3		notnot 282	. . . 4 ⊢ (ψ ↔ ¬ ¬ ψ)
4	3	anbi2i 675	. . 3 ⊢ ((φ ∧ ψ) ↔ (φ ∧ ¬ ¬ ψ))
5		ancom 437	. . 3 ⊢ ((¬ φ ∧ ¬ ψ) ↔ (¬ ψ ∧ ¬ φ))
6	4, 5	orbi12i 507	. 2 ⊢ (((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)) ↔ ((φ ∧ ¬ ¬ ψ) ∨ (¬ ψ ∧ ¬ φ)))
7	1, 2, 6	3bitr4i 268	1 ⊢ ((φ ↔ ψ) ↔ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)))

Syntax hints:  ¬ wn 3   ↔ 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:  pm5.24  864  4exmid  905  nanbi  1294  ifbi  3680