Theorem rexlimdv3d 43647

Description: An extended version of rexlimdvv 3219 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.)

Hypothesis

Ref	Expression
rexlimdv3d.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒)))

Assertion

Ref	Expression
rexlimdv3d	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒))

Distinct variable groups:   𝑧,𝐵   𝑦,𝐴,𝑧   𝜑,𝑥,𝑦,𝑧   𝜒,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜓(𝑥, 𝑦, 𝑧)   𝐴(𝑥)   𝐵(𝑥, 𝑦)   𝐶(𝑥, 𝑦, 𝑧)

Proof of Theorem rexlimdv3d

Step	Hyp	Ref	Expression
1		rexlimdv3d.1	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒)))
2	1	3expd 1372	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑧 ∈ 𝐶 → (𝜓 → 𝜒)))))
3	2	imp4d 430	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → (𝜓 → 𝜒)))
4	3	expdimp 458	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒)))
5	4	rexlimdvv 3219	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒))
6	5	rexlimdva 3164	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145  ∃wrex 3087

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

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-rex 3088

This theorem is used by:  3cubes  43654