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Theorem tfis3 4272
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 4271 . 2 (𝑥 ∈ On → 𝜑)
51, 4vtoclga 2616 1 (𝐴 ∈ On → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393  wral 2303  Oncon0 4072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4232
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-in 2921  df-ss 2928  df-uni 3578  df-tr 3852  df-iord 4075  df-on 4077
This theorem is referenced by:  tfisi  4273  tfrlemi1  5909  rdgon  5936
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