Theorem 3orim123da 1471

Description: Disjoin antecedents and consequents of three premises. (Contributed by Thierry Arnoux, 13-Jul-2026.)

Hypotheses

Ref	Expression
3orim123da.1	⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜂))
3orim123da.2	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3orim123da.3	⊢ ((𝜑 ∧ 𝜃) → 𝜏)
3orim123da.4	⊢ ((𝜑 ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
3orim123da	⊢ (𝜑 → (𝜒 ∨ 𝜏 ∨ 𝜁))

Proof of Theorem 3orim123da

Step	Hyp	Ref	Expression
1		3orim123da.1	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜂))
2		3orim123da.2	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3	2	ex 417	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
4		3orim123da.3	. . . 4 ⊢ ((𝜑 ∧ 𝜃) → 𝜏)
5	4	ex 417	. . 3 ⊢ (𝜑 → (𝜃 → 𝜏))
6		3orim123da.4	. . . 4 ⊢ ((𝜑 ∧ 𝜂) → 𝜁)
7	6	ex 417	. . 3 ⊢ (𝜑 → (𝜂 → 𝜁))
8	3, 5, 7	3orim123d 1470	. 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) → (𝜒 ∨ 𝜏 ∨ 𝜁)))
9	1, 8	mpd 16	1 ⊢ (𝜑 → (𝜒 ∨ 𝜏 ∨ 𝜁))

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102

This theorem is referenced by:  mirlni  28951