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Metamath Proof Explorer

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 ) ) )