Theorem iscnrm3lem7 49666

Description: Lemma for iscnrm3rlem8 49674 and iscnrm3llem2 49677 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 3602	. . 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 49665	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ∃wrex 3096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097

This theorem is referenced by:  iscnrm3rlem8  49674  iscnrm3llem2  49677