Theorem orbidi

Description: Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384 . (Contributed by NM, 8-Jan-2005) (Proof shortened by Wolf Lammen, 4-Feb-2013)

	Ref	Expression
Assertion	orbidi	⊢ ( ( 𝜑 ∨ ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.74	⊢ ( ( ¬ 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( ¬ 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 → 𝜒 ) ) )
2		df-or	⊢ ( ( 𝜑 ∨ ( 𝜓 ↔ 𝜒 ) ) ↔ ( ¬ 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )
3		df-or	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 → 𝜓 ) )
4		df-or	⊢ ( ( 𝜑 ∨ 𝜒 ) ↔ ( ¬ 𝜑 → 𝜒 ) )
5	3 4	bibi12i	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ 𝜒 ) ) ↔ ( ( ¬ 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 → 𝜒 ) ) )
6	1 2 5	3bitr4i	⊢ ( ( 𝜑 ∨ ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ 𝜒 ) ) )