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Theorem mnuss2d 44260
Description: mnussd 44259 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 480 . . 3 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝑈𝑀)
5 simprl 771 . . 3 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝑥𝑈)
6 simprr 773 . . 3 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝐴𝑥)
72, 4, 5, 6mnussd 44259 . 2 ((𝜑 ∧ (𝑥𝑈𝐴𝑥)) → 𝐴𝑈)
81, 7rexlimddv 3159 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963  𝒫 cpw 4605   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-uni 4913
This theorem is referenced by:  mnupwd  44263  mnuunid  44273  mnurndlem2  44278
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