Theorem 3adantll2

Description: Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypothesis	3adantll2.1	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
Assertion	3adantll2	⊢ ( ( ( ( 𝜑 ∧ 𝜂 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		3adantll2.1	⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )
2		simpll1	⊢ ( ( ( ( 𝜑 ∧ 𝜂 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜑 )
3		simpll3	⊢ ( ( ( ( 𝜑 ∧ 𝜂 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜓 )
4	2 3	jca	⊢ ( ( ( ( 𝜑 ∧ 𝜂 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → ( 𝜑 ∧ 𝜓 ) )
5		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝜂 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜒 )
6		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝜂 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜃 )
7	4 5 6 1	syl21anc	⊢ ( ( ( ( 𝜑 ∧ 𝜂 ∧ 𝜓 ) ∧ 𝜒 ) ∧ 𝜃 ) → 𝜏 )