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Theorem rexlimdv3d 39795
 Description: An extended version of rexlimdvv 3253 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.)
Hypothesis
Ref Expression
rexlimdv3d.1 (𝜑 → ((𝑥𝐴𝑦𝐵𝑧𝐶) → (𝜓𝜒)))
Assertion
Ref Expression
rexlimdv3d (𝜑 → (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜓𝜒))
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
StepHypRef Expression
1 rexlimdv3d.1 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦𝐵𝑧𝐶) → (𝜓𝜒)))
213expd 1350 . . . . 5 (𝜑 → (𝑥𝐴 → (𝑦𝐵 → (𝑧𝐶 → (𝜓𝜒)))))
32imp4d 428 . . . 4 (𝜑 → ((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → (𝜓𝜒)))
43expdimp 456 . . 3 ((𝜑𝑥𝐴) → ((𝑦𝐵𝑧𝐶) → (𝜓𝜒)))
54rexlimdvv 3253 . 2 ((𝜑𝑥𝐴) → (∃𝑦𝐵𝑧𝐶 𝜓𝜒))
65rexlimdva 3244 1 (𝜑 → (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-ral 3111  df-rex 3112 This theorem is referenced by:  3cubes  39802
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