MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfis2 Unicode version

Theorem tfis2 4684
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
Hypotheses
Ref Expression
tfis2.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
tfis2.2  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
Assertion
Ref Expression
tfis2  |-  ( x  e.  On  ->  ph )
Distinct variable groups:    ps, x    ph, y    x, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem tfis2
StepHypRef Expression
1 nfv 1610 . 2  |-  F/ x ps
2 tfis2.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3 tfis2.2 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
41, 2, 3tfis2f 4683 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1701   A.wral 2577   Oncon0 4429
This theorem is referenced by:  tfis3  4685  smogt  6426  tfrlem1  6433  findcard3  7145  ordiso2  7275  cantnf  7440  cfsmolem  7941  fpwwe2lem8  8304  nqereu  8598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433
  Copyright terms: Public domain W3C validator