Theorem pm2.82

Description: Theorem *2.82 of WhiteheadRussell p. 108. (Contributed by NM, 3-Jan-2005)

	Ref	Expression
Assertion	pm2.82	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) → ( ( ( 𝜑 ∨ ¬ 𝜒 ) ∨ 𝜃 ) → ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.24	⊢ ( 𝜒 → ( ¬ 𝜒 → 𝜓 ) )
2	1	orim2d	⊢ ( 𝜒 → ( ( 𝜑 ∨ ¬ 𝜒 ) → ( 𝜑 ∨ 𝜓 ) ) )
3	2	jao1i	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) → ( ( 𝜑 ∨ ¬ 𝜒 ) → ( 𝜑 ∨ 𝜓 ) ) )
4	3	orim1d	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) → ( ( ( 𝜑 ∨ ¬ 𝜒 ) ∨ 𝜃 ) → ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜃 ) ) )