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Theorem tfis2f 4234
 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 → (y x 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 1656 . . . 4 ([y / x]φψ)
43ralbii 2308 . . 3 (y x [y / x]φy x ψ)
5 tfis2f.3 . . 3 (x On → (y x ψφ))
64, 5syl5bi 141 . 2 (x On → (y x [y / x]φφ))
76tfis 4233 1 (x On → φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1329   ∈ wcel 1374  [wsb 1627  ∀wral 2284  Oncon0 4049 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054 This theorem is referenced by:  tfis2  4235  tfri3  5875
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