MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfis2f Structured version   Unicode version

Theorem tfis2f 4835
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
Hypotheses
Ref Expression
tfis2f.1  |-  F/ x ps
tfis2f.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
tfis2f.3  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
Assertion
Ref Expression
tfis2f  |-  ( x  e.  On  ->  ph )
Distinct variable groups:    ph, y    x, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem tfis2f
StepHypRef Expression
1 tfis2f.1 . . . . 5  |-  F/ x ps
2 tfis2f.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
31, 2sbie 2149 . . . 4  |-  ( [ y  /  x ] ph 
<->  ps )
43ralbii 2729 . . 3  |-  ( A. y  e.  x  [
y  /  x ] ph 
<-> 
A. y  e.  x  ps )
5 tfis2f.3 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
64, 5syl5bi 209 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph ) )
76tfis 4834 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1553   [wsb 1658    e. wcel 1725   A.wral 2705   Oncon0 4581
This theorem is referenced by:  tfis2  4836  tfr3  6660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
  Copyright terms: Public domain W3C validator