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Theorem tfis2f 4353
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
Hypotheses
Ref Expression
tfis2f.1 𝑥𝜓
tfis2f.2 (𝑥 = 𝑦 → (𝜑𝜓))
tfis2f.3 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
Assertion
Ref Expression
tfis2f (𝑥 ∈ On → 𝜑)
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem tfis2f
StepHypRef Expression
1 tfis2f.1 . . . . 5 𝑥𝜓
2 tfis2f.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2sbie 1716 . . . 4 ([𝑦 / 𝑥]𝜑𝜓)
43ralbii 2377 . . 3 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
5 tfis2f.3 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
64, 5syl5bi 150 . 2 (𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))
76tfis 4352 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wnf 1390  wcel 1434  [wsb 1687  wral 2353  Oncon0 4146
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 2065  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-in 2988  df-ss 2995  df-uni 3622  df-tr 3896  df-iord 4149  df-on 4151
This theorem is referenced by:  tfis2  4354  tfri3  6037
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