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Theorem tfis2f 7797
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, 2sbiev 2309 . . . 4 ([𝑦 / 𝑥]𝜑𝜓)
43ralbii 3097 . . 3 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
5 tfis2f.3 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
64, 5biimtrid 241 . 2 (𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))
76tfis 7796 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wnf 1786  [wsb 2068  wcel 2107  wral 3065  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  tfis2  7798  tfr3  8350
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