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Theorem tfindes 4652
Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction hypothesis for successors, and the third is the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 5-Mar-2004.)
Hypotheses
Ref Expression
tfindes.1  |-  [. (/)  /  x ]. ph
tfindes.2  |-  ( x  e.  On  ->  ( ph  ->  [. suc  x  /  x ]. ph ) )
tfindes.3  |-  ( Lim  y  ->  ( A. x  e.  y  ph  ->  [. y  /  x ]. ph ) )
Assertion
Ref Expression
tfindes  |-  ( x  e.  On  ->  ph )
Distinct variable groups:    x, y    ph, y
Dummy variable  z is distinct from all other variables.
Allowed substitution hint:    ph( x)

Proof of Theorem tfindes
StepHypRef Expression
1 dfsbcq 2994 . 2  |-  ( y  =  (/)  ->  ( [. y  /  x ]. ph  <->  [. (/)  /  x ]. ph ) )
2 dfsbcq 2994 . 2  |-  ( y  =  z  ->  ( [. y  /  x ]. ph  <->  [. z  /  x ]. ph ) )
3 dfsbcq 2994 . 2  |-  ( y  =  suc  z  -> 
( [. y  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
4 sbceq2a 3003 . 2  |-  ( y  =  x  ->  ( [. y  /  x ]. ph  <->  ph ) )
5 tfindes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1606 . . . 4  |-  F/ x  z  e.  On
7 nfsbc1v 3011 . . . . 5  |-  F/ x [. z  /  x ]. ph
8 nfsbc1v 3011 . . . . 5  |-  F/ x [. suc  z  /  x ]. ph
97, 8nfim 1770 . . . 4  |-  F/ x
( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
106, 9nfim 1770 . . 3  |-  F/ x
( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
11 eleq1 2344 . . . 4  |-  ( x  =  z  ->  (
x  e.  On  <->  z  e.  On ) )
12 sbceq1a 3002 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
13 suceq 4456 . . . . . 6  |-  ( x  =  z  ->  suc  x  =  suc  z )
14 dfsbcq 2994 . . . . . 6  |-  ( suc  x  =  suc  z  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  z  /  x ]. ph )
)
1513, 14syl 17 . . . . 5  |-  ( x  =  z  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
1612, 15imbi12d 313 . . . 4  |-  ( x  =  z  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  (
[. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
) )
1711, 16imbi12d 313 . . 3  |-  ( x  =  z  ->  (
( x  e.  On  ->  ( ph  ->  [. suc  x  /  x ]. ph )
)  <->  ( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [.
suc  z  /  x ]. ph ) ) ) )
18 tfindes.2 . . 3  |-  ( x  e.  On  ->  ( ph  ->  [. suc  x  /  x ]. ph ) )
1910, 17, 18chvar 1928 . 2  |-  ( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
20 cbvralsv 2776 . . . 4  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [
z  /  x ] ph )
21 sbsbc 2996 . . . . 5  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
2221ralbii 2568 . . . 4  |-  ( A. z  e.  y  [
z  /  x ] ph 
<-> 
A. z  e.  y 
[. z  /  x ]. ph )
2320, 22bitri 242 . . 3  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [. z  /  x ]. ph )
24 tfindes.3 . . 3  |-  ( Lim  y  ->  ( A. x  e.  y  ph  ->  [. y  /  x ]. ph ) )
2523, 24syl5bir 211 . 2  |-  ( Lim  y  ->  ( A. z  e.  y  [. z  /  x ]. ph  ->  [. y  /  x ]. ph ) )
261, 2, 3, 4, 5, 19, 25tfinds 4649 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1624   [wsb 1631    e. wcel 1685   A.wral 2544   [.wsbc 2992   (/)c0 3456   Oncon0 4391   Lim wlim 4392   suc csuc 4393
This theorem is referenced by:  tfinds2  4653
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397
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