Theorem iscnrm3lem5

Description: Lemma for iscnrm3l . (Contributed by Zhi Wang, 3-Sep-2024)

	Ref	Expression
Hypotheses	iscnrm3lem5.1	⊢ ( ( 𝑥 = 𝑆 ∧ 𝑦 = 𝑇 ) → ( 𝜑 ↔ 𝜓 ) )
	iscnrm3lem5.2	⊢ ( ( 𝑥 = 𝑆 ∧ 𝑦 = 𝑇 ) → ( 𝜒 ↔ 𝜃 ) )
	iscnrm3lem5.3	⊢ ( ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( 𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊 ) )
	iscnrm3lem5.4	⊢ ( ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( ( 𝜓 → 𝜃 ) → 𝜎 ) )
Assertion	iscnrm3lem5	⊢ ( 𝜏 → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑊 ( 𝜑 → 𝜒 ) → ( 𝜂 → ( 𝜁 → 𝜎 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		iscnrm3lem5.1	⊢ ( ( 𝑥 = 𝑆 ∧ 𝑦 = 𝑇 ) → ( 𝜑 ↔ 𝜓 ) )
2		iscnrm3lem5.2	⊢ ( ( 𝑥 = 𝑆 ∧ 𝑦 = 𝑇 ) → ( 𝜒 ↔ 𝜃 ) )
3		iscnrm3lem5.3	⊢ ( ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( 𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊 ) )
4		iscnrm3lem5.4	⊢ ( ( 𝜏 ∧ 𝜂 ∧ 𝜁 ) → ( ( 𝜓 → 𝜃 ) → 𝜎 ) )
5	1 2	imbi12d	⊢ ( ( 𝑥 = 𝑆 ∧ 𝑦 = 𝑇 ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) )
6	5	rspc2gv	⊢ ( ( 𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊 ) → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑊 ( 𝜑 → 𝜒 ) → ( 𝜓 → 𝜃 ) ) )
7	6 3 4	iscnrm3lem4	⊢ ( 𝜏 → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑊 ( 𝜑 → 𝜒 ) → ( 𝜂 → ( 𝜁 → 𝜎 ) ) ) )