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Theorem tfindsd 44229
Description: Deduction associated with tfinds 7882. (Contributed by Rohan Ridenour, 8-Aug-2023.)
Hypotheses
Ref Expression
tfindsd.1 (𝑥 = ∅ → (𝜓𝜒))
tfindsd.2 (𝑥 = 𝑦 → (𝜓𝜃))
tfindsd.3 (𝑥 = suc 𝑦 → (𝜓𝜏))
tfindsd.4 (𝑥 = 𝐴 → (𝜓𝜂))
tfindsd.5 (𝜑𝜒)
tfindsd.6 ((𝜑𝑦 ∈ On ∧ 𝜃) → 𝜏)
tfindsd.7 ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦𝑥 𝜃) → 𝜓)
tfindsd.8 (𝜑𝐴 ∈ On)
Assertion
Ref Expression
tfindsd (𝜑𝜂)
Distinct variable groups:   𝜓,𝑦   𝜃,𝑥   𝜂,𝑥   𝑥,𝐴   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥,𝑦)   𝜃(𝑦)   𝜏(𝑥,𝑦)   𝜂(𝑦)   𝐴(𝑦)

Proof of Theorem tfindsd
StepHypRef Expression
1 tfindsd.8 . 2 (𝜑𝐴 ∈ On)
2 tfindsd.1 . . 3 (𝑥 = ∅ → (𝜓𝜒))
3 tfindsd.2 . . 3 (𝑥 = 𝑦 → (𝜓𝜃))
4 tfindsd.3 . . 3 (𝑥 = suc 𝑦 → (𝜓𝜏))
5 tfindsd.4 . . 3 (𝑥 = 𝐴 → (𝜓𝜂))
6 tfindsd.5 . . 3 (𝜑𝜒)
7 tfindsd.6 . . . . 5 ((𝜑𝑦 ∈ On ∧ 𝜃) → 𝜏)
873exp 1119 . . . 4 (𝜑 → (𝑦 ∈ On → (𝜃𝜏)))
98com12 32 . . 3 (𝑦 ∈ On → (𝜑 → (𝜃𝜏)))
10 tfindsd.7 . . . . 5 ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦𝑥 𝜃) → 𝜓)
11103exp 1119 . . . 4 (𝜑 → (Lim 𝑥 → (∀𝑦𝑥 𝜃𝜓)))
1211com12 32 . . 3 (Lim 𝑥 → (𝜑 → (∀𝑦𝑥 𝜃𝜓)))
132, 3, 4, 5, 6, 9, 12tfinds3 7887 . 2 (𝐴 ∈ On → (𝜑𝜂))
141, 13mpcom 38 1 (𝜑𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1539  wcel 2107  wral 3060  c0 4332  Oncon0 6383  Lim wlim 6384  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389
This theorem is referenced by:  grur1cld  44256
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