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Theorem tfis2 7290
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 2010 . 2 𝑥𝜓
2 tfis2.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
3 tfis2.2 . 2 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
41, 2, 3tfis2f 7289 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2157  wral 3089  Oncon0 5941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945
This theorem is referenced by:  tfis3  7291  smogt  7703  findcard3  8445  ordiso2  8662  cantnf  8840  cfsmolem  9380  fpwwe2lem8  9747  nqereu  10039  tfis2d  43226
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