Theorem iscnrm3lem7 49745

Description: Lemma for iscnrm3rlem8 49753 and iscnrm3llem2 49756 involving restricted existential quantifications. (Contributed by Zhi Wang, 5-Sep-2024.)

Hypotheses

Ref	Expression
iscnrm3lem7.1	⊢ (𝑧 = 𝑍 → (𝜒 ↔ 𝜃))
iscnrm3lem7.2	⊢ (𝑤 = 𝑊 → (𝜃 ↔ 𝜏))
iscnrm3lem7.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏))

Assertion

Ref	Expression
iscnrm3lem7	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒))

Distinct variable groups:   𝑦,𝐴   𝑥,𝐶,𝑦,𝑧   𝑤,𝐷,𝑥,𝑦,𝑧   𝑤,𝑊   𝑤,𝑍,𝑧   𝜒,𝑥,𝑦   𝜑,𝑥,𝑦   𝜏,𝑤   𝜃,𝑧

Allowed substitution hints:   𝜑(𝑧, 𝑤)   𝜓(𝑥, 𝑦, 𝑧, 𝑤)   𝜒(𝑧, 𝑤)   𝜃(𝑥, 𝑦, 𝑤)   𝜏(𝑥, 𝑦, 𝑧)   𝐴(𝑥, 𝑧, 𝑤)   𝐵(𝑥, 𝑦, 𝑧, 𝑤)   𝐶(𝑤)   𝑊(𝑥, 𝑦, 𝑧)   𝑍(𝑥, 𝑦)

Proof of Theorem iscnrm3lem7

Step	Hyp	Ref	Expression
1		iscnrm3lem7.3	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏))
2		iscnrm3lem7.1	. . . 4 ⊢ (𝑧 = 𝑍 → (𝜒 ↔ 𝜃))
3		iscnrm3lem7.2	. . . 4 ⊢ (𝑤 = 𝑊 → (𝜃 ↔ 𝜏))
4	2, 3	rspc2ev 3593	. . 3 ⊢ ((𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏) → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒)
5	1, 4	syl 18	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒)
6	5	iscnrm3lem6 49744	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089

This theorem is used by:  iscnrm3rlem8  49753  iscnrm3llem2  49756