Theorem rexlimdv3d 42694

Description: An extended version of rexlimdvv 3212 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 1354	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑧 ∈ 𝐶 → (𝜓 → 𝜒)))))
3	2	imp4d 424	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → (𝜓 → 𝜒)))
4	3	expdimp 452	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒)))
5	4	rexlimdvv 3212	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒))
6	5	rexlimdva 3155	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1780  df-rex 3071

This theorem is referenced by:  3cubes  42701