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Theorem tfis3 6949
 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 6948 . 2 (𝑥 ∈ On → 𝜑)
51, 4vtoclga 3245 1 (𝐴 ∈ On → 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by:  tfisi  6950  tfinds  6951  tfrlem1  7359  ordtypelem7  8312  rankonidlem  8574  tcrank  8630  infxpenlem  8719  alephle  8794  dfac12lem3  8850  ttukeylem5  9218  ttukeylem6  9219  tskord  9481  grudomon  9518  aomclem6  36647
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