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Theorem tfindes 4842
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
Allowed substitution hint:    ph( x)

Proof of Theorem tfindes
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3163 . 2  |-  ( y  =  (/)  ->  ( [. y  /  x ]. ph  <->  [. (/)  /  x ]. ph ) )
2 dfsbcq 3163 . 2  |-  ( y  =  z  ->  ( [. y  /  x ]. ph  <->  [. z  /  x ]. ph ) )
3 dfsbcq 3163 . 2  |-  ( y  =  suc  z  -> 
( [. y  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
4 sbceq2a 3172 . 2  |-  ( y  =  x  ->  ( [. y  /  x ]. ph  <->  ph ) )
5 tfindes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1629 . . . 4  |-  F/ x  z  e.  On
7 nfsbc1v 3180 . . . . 5  |-  F/ x [. z  /  x ]. ph
8 nfsbc1v 3180 . . . . 5  |-  F/ x [. suc  z  /  x ]. ph
97, 8nfim 1832 . . . 4  |-  F/ x
( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
106, 9nfim 1832 . . 3  |-  F/ x
( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
11 eleq1 2496 . . . 4  |-  ( x  =  z  ->  (
x  e.  On  <->  z  e.  On ) )
12 sbceq1a 3171 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
13 suceq 4646 . . . . . 6  |-  ( x  =  z  ->  suc  x  =  suc  z )
14 dfsbcq 3163 . . . . . 6  |-  ( suc  x  =  suc  z  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  z  /  x ]. ph )
)
1513, 14syl 16 . . . . 5  |-  ( x  =  z  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
1612, 15imbi12d 312 . . . 4  |-  ( x  =  z  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  (
[. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
) )
1711, 16imbi12d 312 . . 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 1968 . 2  |-  ( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
20 cbvralsv 2943 . . . 4  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [
z  /  x ] ph )
21 sbsbc 3165 . . . . 5  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
2221ralbii 2729 . . . 4  |-  ( A. z  e.  y  [
z  /  x ] ph 
<-> 
A. z  e.  y 
[. z  /  x ]. ph )
2320, 22bitri 241 . . 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 210 . 2  |-  ( Lim  y  ->  ( A. z  e.  y  [. z  /  x ]. ph  ->  [. y  /  x ]. ph ) )
261, 2, 3, 4, 5, 19, 25tfinds 4839 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   [wsb 1658    e. wcel 1725   A.wral 2705   [.wsbc 3161   (/)c0 3628   Oncon0 4581   Lim wlim 4582   suc csuc 4583
This theorem is referenced by:  tfinds2  4843
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-pw 3801  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  df-lim 4586  df-suc 4587
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