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Theorem tfis2f 4595
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 1801 . . . 4 ([𝑦 / 𝑥]𝜑𝜓)
43ralbii 2493 . . 3 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
5 tfis2f.3 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
64, 5biimtrid 152 . 2 (𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))
76tfis 4594 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wnf 1470  [wsb 1772  wcel 2158  wral 2465  Oncon0 4375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-in 3147  df-ss 3154  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380
This theorem is referenced by:  tfis2  4596  tfri3  6381
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