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Theorem tfindsd 40852
 Description: Deduction associated with tfinds 7559. (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 1116 . . . 4 (𝜑 → (𝑦 ∈ On → (𝜃𝜏)))
98com12 32 . . 3 (𝑦 ∈ On → (𝜑 → (𝜃𝜏)))
10 tfindsd.7 . . . . 5 ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦𝑥 𝜃) → 𝜓)
11103exp 1116 . . . 4 (𝜑 → (Lim 𝑥 → (∀𝑦𝑥 𝜃𝜓)))
1211com12 32 . . 3 (Lim 𝑥 → (𝜑 → (∀𝑦𝑥 𝜃𝜓)))
132, 3, 4, 5, 6, 9, 12tfinds3 7564 . 2 (𝐴 ∈ On → (𝜑𝜂))
141, 13mpcom 38 1 (𝜑𝜂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ∅c0 4265  Oncon0 6169  Lim wlim 6170  suc csuc 6171 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175 This theorem is referenced by:  grur1cld  40874
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