Theorem 3imp3i2an

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017) (Proof shortened by Wolf Lammen, 13-Apr-2022)

	Ref	Expression
Hypotheses	3imp3i2an.1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
	3imp3i2an.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜏 )
	3imp3i2an.3	⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
Assertion	3imp3i2an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		3imp3i2an.1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
2		3imp3i2an.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜏 )
3		3imp3i2an.3	⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
4	2	3adant2	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜏 )
5	1 4 3	syl2anc	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜂 )