Theorem iscnrm3lem6 49744

Description: Lemma for iscnrm3lem7 49745. (Contributed by Zhi Wang, 5-Sep-2024.)

Hypothesis

Ref	Expression
iscnrm3lem6.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
iscnrm3lem6	⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))

Distinct variable groups:   𝑦,𝑉   𝜒,𝑥,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥, 𝑦)   𝑉(𝑥)   𝑊(𝑥, 𝑦)

Proof of Theorem iscnrm3lem6

Step	Hyp	Ref	Expression
1		iscnrm3lem6.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)
2	1	3exp 1136	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) → (𝜓 → 𝜒)))
3	2	rexlimdvv 3220	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   ∈ 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

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-ex 1809  df-rex 3089

This theorem is used by:  iscnrm3lem7  49745