Theorem iscnrm3lem5 50044

Description: Lemma for iscnrm3l 50058. (Contributed by Zhi Wang, 3-Sep-2024.)

Hypotheses

Ref	Expression
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

Ref	Expression
iscnrm3lem5	⊢ (𝜏 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝜑 → 𝜒) → (𝜂 → (𝜁 → 𝜎))))

Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝜓,𝑥,𝑦   𝜃,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜒(𝑥, 𝑦)   𝜏(𝑥, 𝑦)   𝜂(𝑥, 𝑦)   𝜁(𝑥, 𝑦)   𝜎(𝑥, 𝑦)

Proof of Theorem iscnrm3lem5

Step	Hyp	Ref	Expression
1		iscnrm3lem5.1	. . . 4 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜑 ↔ 𝜓))
2		iscnrm3lem5.2	. . . 4 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜒 ↔ 𝜃))
3	1, 2	imbi12d 347	. . 3 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
4	3	rspc2gv 3586	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝜑 → 𝜒) → (𝜓 → 𝜃)))
5		iscnrm3lem5.3	. 2 ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜁) → (𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊))
6		iscnrm3lem5.4	. 2 ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝜓 → 𝜃) → 𝜎))
7	4, 5, 6	iscnrm3lem4 50043	1 ⊢ (𝜏 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝜑 → 𝜒) → (𝜂 → (𝜁 → 𝜎))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078

This theorem is used by:  iscnrm3l  50058