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Theorem tfis3 4329
 Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
Hypotheses
Ref Expression
tfis3.1 (𝑥 = 𝑦 → (𝜑𝜓))
tfis3.2 (𝑥 = 𝐴 → (𝜑𝜒))
tfis3.3 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
Assertion
Ref Expression
tfis3 (𝐴 ∈ On → 𝜒)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝜒,𝑥   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem tfis3
StepHypRef Expression
1 tfis3.2 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
2 tfis3.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
3 tfis3.3 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
42, 3tfis2 4328 . 2 (𝑥 ∈ On → 𝜑)
51, 4vtoclga 2665 1 (𝐴 ∈ On → 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   = wceq 1285   ∈ wcel 1434  ∀wral 2349  Oncon0 4120 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4282 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-in 2980  df-ss 2987  df-uni 3604  df-tr 3878  df-iord 4123  df-on 4125 This theorem is referenced by:  tfisi  4330  tfrlemi1  5975  tfr1onlemaccex  5991  tfrcllemaccex  6004  tfrcl  6007
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