Theorem iscnrm3lem4

Description: Lemma for iscnrm3lem5 and iscnrm3r . (Contributed by Zhi Wang, 4-Sep-2024)

	Ref	Expression
Hypotheses	iscnrm3lem4.1	⊢ ( 𝜂 → ( 𝜓 → 𝜁 ) )
	iscnrm3lem4.2	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜂 )
	iscnrm3lem4.3	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → ( 𝜁 → 𝜏 ) )
Assertion	iscnrm3lem4	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		iscnrm3lem4.1	⊢ ( 𝜂 → ( 𝜓 → 𝜁 ) )
2		iscnrm3lem4.2	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜂 )
3		iscnrm3lem4.3	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → ( 𝜁 → 𝜏 ) )
4		4anpull2	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜓 ) )
5	2 1	syl	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → ( 𝜓 → 𝜁 ) )
6	5 3	syld	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → ( 𝜓 → 𝜏 ) )
7	6	imp	⊢ ( ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) ∧ 𝜓 ) → 𝜏 )
8	4 7	sylbi	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) → 𝜏 )
9	8	exp43	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) ) )