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Theorem tfis2 7845
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
Hypotheses
Ref Expression
tfis2.1 (𝑥 = 𝑦 → (𝜑𝜓))
tfis2.2 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
Assertion
Ref Expression
tfis2 (𝑥 ∈ On → 𝜑)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem tfis2
StepHypRef Expression
1 nfv 1917 . 2 𝑥𝜓
2 tfis2.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
3 tfis2.2 . 2 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
41, 2, 3tfis2f 7844 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  wral 3061  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  tfis3  7846  smogt  8366  findcard3  9284  findcard3OLD  9285  ordiso2  9509  cantnf  9687  cfsmolem  10264  fpwwe2lem7  10631  nqereu  10923  addsprop  27457  negsprop  27506  mulsprop  27583  tfis2d  47715
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