Theorem iscnrm3lem2 45667

Description: Lemma for iscnrm3 45685 proving a biconditional on restricted universal quantifications. (Contributed by Zhi Wang, 3-Sep-2024.)

Hypotheses

Ref	Expression
iscnrm3lem2.1	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 → ((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))
iscnrm3lem2.2	⊢ (𝜑 → (∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))

Assertion

Ref	Expression
iscnrm3lem2	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒))

Distinct variable groups:   𝑣,𝐴,𝑤,𝑦,𝑧   𝑣,𝐵,𝑤,𝑧   𝑣,𝐶,𝑤   𝑣,𝐷,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝜒,𝑥,𝑦,𝑧   𝜑,𝑣,𝑤,𝑥,𝑦,𝑧   𝜓,𝑣,𝑤

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑤,𝑣)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧)   𝐷(𝑤)   𝐸(𝑤,𝑣)

Proof of Theorem iscnrm3lem2

Step	Hyp	Ref	Expression
1		2ax5 1938	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓) → ∀𝑤∀𝑣∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓))
2		r3al 3131	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓))
3		iscnrm3lem2.1	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 → ((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))
4	2, 3	syl5bir 246	. . . . 5 ⊢ (𝜑 → (∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓) → ((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))
5	4	2alimdv 1919	. . . 4 ⊢ (𝜑 → (∀𝑤∀𝑣∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓) → ∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))
6	1, 5	syl5 34	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓) → ∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))
7		2ax5 1938	. . . . 5 ⊢ (∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒) → ∀𝑦∀𝑧∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒))
8	7	alrimiv 1928	. . . 4 ⊢ (∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒) → ∀𝑥∀𝑦∀𝑧∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒))
9		r2al 3130	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒 ↔ ∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒))
10		iscnrm3lem2.2	. . . . . . 7 ⊢ (𝜑 → (∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))
11	9, 10	syl5bir 246	. . . . . 6 ⊢ (𝜑 → (∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))
12	11	2alimdv 1919	. . . . 5 ⊢ (𝜑 → (∀𝑦∀𝑧∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒) → ∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))
13	12	alimdv 1917	. . . 4 ⊢ (𝜑 → (∀𝑥∀𝑦∀𝑧∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒) → ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))
14	8, 13	syl5 34	. . 3 ⊢ (𝜑 → (∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒) → ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))
15	6, 14	impbid 215	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓) ↔ ∀𝑤∀𝑣((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))
16	15, 2, 9	3bitr4g 317	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   ∈ wcel 2111  ∀wral 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-ral 3075

This theorem is referenced by:  iscnrm3  45685