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Theorem axprOLD 5436
Description: Obsolete version of axpr 5432 as of 18-Sep-2025. (Contributed by NM, 14-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axprOLD 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤

Proof of Theorem axprOLD
Dummy variables 𝑠 𝑝 𝑡 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axprlem3OLD 5433 . . . 4 𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
2 biimpr 220 . . . . 5 ((𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → (∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧))
32alimi 1807 . . . 4 (∀𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) → ∀𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧))
41, 3eximii 1833 . . 3 𝑧𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧)
5 axprlem4OLD 5434 . . . . . . . 8 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
6 axprlem5OLD 5435 . . . . . . . 8 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
75, 6jaodan 959 . . . . . . 7 ((∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) ∧ (𝑤 = 𝑥𝑤 = 𝑦)) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
87ex 412 . . . . . 6 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ((𝑤 = 𝑥𝑤 = 𝑦) → ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
98imim1d 82 . . . . 5 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ((∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧) → ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
109alimdv 1913 . . . 4 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (∀𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧) → ∀𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
1110eximdv 1914 . . 3 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → (∃𝑧𝑤(∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)) → 𝑤𝑧) → ∃𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)))
124, 11mpi 20 . 2 (∀𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝) → ∃𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
13 axprlem2 5429 . 2 𝑝𝑠(∀𝑛𝑠𝑡 ¬ 𝑡𝑛𝑠𝑝)
1412, 13exlimiiv 1928 1 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  if-wif 1062  wal 1534  wex 1775  wral 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-10 2138  ax-12 2174  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-tru 1539  df-ex 1776  df-nf 1780  df-ral 3059
This theorem is referenced by: (None)
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