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Theorem axprlem5OLD 5430
Description: Obsolete version of axprlem4 5426 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axprlem5OLD ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Distinct variable groups:   𝑦,𝑠   𝑤,𝑠   𝑛,𝑠

Proof of Theorem axprlem5OLD
StepHypRef Expression
1 ax-nul 5306 . 2 𝑠𝑛 ¬ 𝑛𝑠
2 nfa1 2151 . . . 4 𝑠𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝)
3 nfv 1914 . . . 4 𝑠 𝑤 = 𝑦
42, 3nfan 1899 . . 3 𝑠(∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)
5 pm2.21 123 . . . . . . . . 9 𝑛𝑠 → (𝑛𝑠 → ∀𝑡 ¬ 𝑡𝑛))
65alimi 1811 . . . . . . . 8 (∀𝑛 ¬ 𝑛𝑠 → ∀𝑛(𝑛𝑠 → ∀𝑡 ¬ 𝑡𝑛))
76adantr 480 . . . . . . 7 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛(𝑛𝑠 → ∀𝑡 ¬ 𝑡𝑛))
8 df-ral 3062 . . . . . . 7 (∀𝑛𝑠𝑡 ¬ 𝑡𝑛 ↔ ∀𝑛(𝑛𝑠 → ∀𝑡 ¬ 𝑡𝑛))
97, 8sylibr 234 . . . . . 6 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛𝑠𝑡 ¬ 𝑡𝑛)
10 sp 2183 . . . . . . 7 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝))
1110ad2antrl 728 . . . . . 6 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → (∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝))
129, 11mpd 15 . . . . 5 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → 𝑠𝑝)
13 simpl 482 . . . . . . 7 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛 ¬ 𝑛𝑠)
14 alnex 1781 . . . . . . 7 (∀𝑛 ¬ 𝑛𝑠 ↔ ¬ ∃𝑛 𝑛𝑠)
1513, 14sylib 218 . . . . . 6 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → ¬ ∃𝑛 𝑛𝑠)
16 simprr 773 . . . . . 6 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑦)
17 ifpfal 1076 . . . . . . 7 (¬ ∃𝑛 𝑛𝑠 → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦))
1817biimpar 477 . . . . . 6 ((¬ ∃𝑛 𝑛𝑠𝑤 = 𝑦) → if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))
1915, 16, 18syl2anc 584 . . . . 5 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))
2012, 19jca 511 . . . 4 ((∀𝑛 ¬ 𝑛𝑠 ∧ (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦)) → (𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
2120expcom 413 . . 3 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦) → (∀𝑛 ¬ 𝑛𝑠 → (𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
224, 21eximd 2216 . 2 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦) → (∃𝑠𝑛 ¬ 𝑛𝑠 → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
231, 22mpi 20 1 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  if-wif 1063  wal 1538  wex 1779  wral 3061
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  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-tru 1543  df-ex 1780  df-nf 1784  df-ral 3062
This theorem is referenced by:  axprOLD  5431
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