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Theorem tfinds3 7563
 Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)
Hypotheses
Ref Expression
tfinds3.1 (𝑥 = ∅ → (𝜑𝜓))
tfinds3.2 (𝑥 = 𝑦 → (𝜑𝜒))
tfinds3.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
tfinds3.4 (𝑥 = 𝐴 → (𝜑𝜏))
tfinds3.5 (𝜂𝜓)
tfinds3.6 (𝑦 ∈ On → (𝜂 → (𝜒𝜃)))
tfinds3.7 (Lim 𝑥 → (𝜂 → (∀𝑦𝑥 𝜒𝜑)))
Assertion
Ref Expression
tfinds3 (𝐴 ∈ On → (𝜂𝜏))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑦   𝜒,𝑥   𝜏,𝑥   𝑥,𝑦,𝜂
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑥,𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem tfinds3
StepHypRef Expression
1 tfinds3.1 . . 3 (𝑥 = ∅ → (𝜑𝜓))
21imbi2d 344 . 2 (𝑥 = ∅ → ((𝜂𝜑) ↔ (𝜂𝜓)))
3 tfinds3.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜒))
43imbi2d 344 . 2 (𝑥 = 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜒)))
5 tfinds3.3 . . 3 (𝑥 = suc 𝑦 → (𝜑𝜃))
65imbi2d 344 . 2 (𝑥 = suc 𝑦 → ((𝜂𝜑) ↔ (𝜂𝜃)))
7 tfinds3.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜏))
87imbi2d 344 . 2 (𝑥 = 𝐴 → ((𝜂𝜑) ↔ (𝜂𝜏)))
9 tfinds3.5 . 2 (𝜂𝜓)
10 tfinds3.6 . . 3 (𝑦 ∈ On → (𝜂 → (𝜒𝜃)))
1110a2d 29 . 2 (𝑦 ∈ On → ((𝜂𝜒) → (𝜂𝜃)))
12 r19.21v 3145 . . 3 (∀𝑦𝑥 (𝜂𝜒) ↔ (𝜂 → ∀𝑦𝑥 𝜒))
13 tfinds3.7 . . . 4 (Lim 𝑥 → (𝜂 → (∀𝑦𝑥 𝜒𝜑)))
1413a2d 29 . . 3 (Lim 𝑥 → ((𝜂 → ∀𝑦𝑥 𝜒) → (𝜂𝜑)))
1512, 14syl5bi 245 . 2 (Lim 𝑥 → (∀𝑦𝑥 (𝜂𝜒) → (𝜂𝜑)))
162, 4, 6, 8, 9, 11, 15tfinds 7558 1 (𝐴 ∈ On → (𝜂𝜏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2112  ∀wral 3109  ∅c0 4246  Oncon0 6163  Lim wlim 6164  suc csuc 6165 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169 This theorem is referenced by:  oacl  8147  omcl  8148  oecl  8149  oawordri  8163  oaass  8174  oarec  8175  omordi  8179  omwordri  8185  odi  8192  omass  8193  oen0  8199  oewordri  8205  oeworde  8206  oeoelem  8211  omabs  8261  tfindsd  40915
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