Theorem 3jaao 1396

Description: Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Hypotheses

Ref	Expression
3jaao.1	⊢ (𝜑 → (𝜓 → 𝜒))
3jaao.2	⊢ (𝜃 → (𝜏 → 𝜒))
3jaao.3	⊢ (𝜂 → (𝜁 → 𝜒))

Assertion

Ref	Expression
3jaao	⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → ((𝜓 ∨ 𝜏 ∨ 𝜁) → 𝜒))

Proof of Theorem 3jaao

Step	Hyp	Ref	Expression
1		3jaao.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	3ad2ant1 1082	. 2 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜓 → 𝜒))
3		3jaao.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜒))
4	3	3ad2ant2 1083	. 2 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜏 → 𝜒))
5		3jaao.3	. . 3 ⊢ (𝜂 → (𝜁 → 𝜒))
6	5	3ad2ant3 1084	. 2 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜁 → 𝜒))
7	2, 4, 6	3jaod 1392	1 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → ((𝜓 ∨ 𝜏 ∨ 𝜁) → 𝜒))

Syntax hints:   → wi 4   ∨ w3o 1036   ∧ w3a 1037

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 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039

This theorem is referenced by:  lpni  27332  3ornot23  38715