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Theorem mnuss2d 42636
Description: mnussd 42635 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 42635 . 2 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝐴𝑈)
81, 7rexlimddv 3155 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  wss 3914  𝒫 cpw 4564   cuni 4869
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 2704  ax-sep 5260
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 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566  df-uni 4870
This theorem is referenced by:  mnupwd  42639  mnuunid  42649  mnurndlem2  42654
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