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Theorem tfindsd 42577
Description: Deduction associated with tfinds 7800. (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 1120 . . . 4 (𝜑 → (𝑦 ∈ On → (𝜃𝜏)))
98com12 32 . . 3 (𝑦 ∈ On → (𝜑 → (𝜃𝜏)))
10 tfindsd.7 . . . . 5 ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦𝑥 𝜃) → 𝜓)
11103exp 1120 . . . 4 (𝜑 → (Lim 𝑥 → (∀𝑦𝑥 𝜃𝜓)))
1211com12 32 . . 3 (Lim 𝑥 → (𝜑 → (∀𝑦𝑥 𝜃𝜓)))
132, 3, 4, 5, 6, 9, 12tfinds3 7805 . 2 (𝐴 ∈ On → (𝜑𝜂))
141, 13mpcom 38 1 (𝜑𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088   = wceq 1542  wcel 2107  wral 3061  c0 4286  Oncon0 6321  Lim wlim 6322  suc csuc 6323
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327
This theorem is referenced by:  grur1cld  42604
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