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Metamath Proof Explorer

Theorem 3jaao

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)

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

Proof

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