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Theorem tfis2 7855
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 1910 . 2 𝑥𝜓
2 tfis2.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
3 tfis2.2 . 2 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
41, 2, 3tfis2f 7854 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2099  wral 3057  Oncon0 6363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367
This theorem is referenced by:  tfis3  7856  smogt  8381  findcard3  9303  findcard3OLD  9304  ordiso2  9532  cantnf  9710  cfsmolem  10287  fpwwe2lem7  10654  nqereu  10946  addsprop  27886  negsprop  27940  mulsprop  28023  tfis2d  48105
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