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Theorem mnuss2d 42345
Description: mnussd 42344 with arguments provided with an existential quantifier. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnuss2d.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnuss2d.2 (𝜑𝑈𝑀)
mnuss2d.3 (𝜑 → ∃𝑥𝑈 𝐴𝑥)
Assertion
Ref Expression
mnuss2d (𝜑𝐴𝑈)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝑈   𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙   𝑈,𝑞,𝑘,𝑚,𝑛,𝑝,𝑙
Allowed substitution hints:   𝜑(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑥,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuss2d
StepHypRef Expression
1 mnuss2d.3 . 2 (𝜑 → ∃𝑥𝑈 𝐴𝑥)
2 mnuss2d.1 . . 3 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
3 mnuss2d.2 . . . 4 (𝜑𝑈𝑀)
43adantr 482 . . 3 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝑈𝑀)
5 simprl 770 . . 3 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝑥𝑈)
6 simprr 772 . . 3 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝐴𝑥)
72, 4, 5, 6mnussd 42344 . 2 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝐴𝑈)
81, 7rexlimddv 3157 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2715  wral 3063  wrex 3072  wss 3909  𝒫 cpw 4559   cuni 4864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-in 3916  df-ss 3926  df-pw 4561  df-uni 4865
This theorem is referenced by:  mnupwd  42348  mnuunid  42358  mnurndlem2  42363
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