This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 3jaobOLD

Description: Obsolete version of 3jaob as of 29-Jun-2025. (Contributed by NM, 13-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 3jaobOLD ( ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 3mix1 ⊢ ( 𝜑 → ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) )
2 1 imim1i ⊢ ( ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) )
3 3mix2 ⊢ ( 𝜒 → ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) )
4 3 imim1i ⊢ ( ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) → ( 𝜒 → 𝜓 ) )
5 3mix3 ⊢ ( 𝜃 → ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) )
6 5 imim1i ⊢ ( ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) → ( 𝜃 → 𝜓 ) )
7 2 4 6 3jca ⊢ ( ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) → ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) ) )
8 3jao ⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) ) → ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) )
9 7 8 impbii ⊢ ( ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) ) )