Metamath Proof Explorer

Theorem pm5.7

Description: Disjunction distributes over the biconditional. Theorem *5.7 of WhiteheadRussell p. 125. This theorem is similar to orbidi . (Contributed by Roy F. Longton, 21-Jun-2005)

Ref Expression
Assertion pm5.7 ⊢ φ ∨ χ ↔ ψ ∨ χ ↔ χ ∨ φ ↔ ψ

Proof

Step Hyp Ref Expression
1 orbidi ⊢ χ ∨ φ ↔ ψ ↔ χ ∨ φ ↔ χ ∨ ψ
2 orcom ⊢ χ ∨ φ ↔ φ ∨ χ
3 orcom ⊢ χ ∨ ψ ↔ ψ ∨ χ
4 2 3 bibi12i ⊢ χ ∨ φ ↔ χ ∨ ψ ↔ φ ∨ χ ↔ ψ ∨ χ
5 1 4 bitr2i ⊢ φ ∨ χ ↔ ψ ∨ χ ↔ χ ∨ φ ↔ ψ