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Theorem tfindes 4625
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
StepHypRef Expression
1 dfsbcq 2968 . 2  |-  ( y  =  (/)  ->  ( [. y  /  x ]. ph  <->  [. (/)  /  x ]. ph ) )
2 dfsbcq 2968 . 2  |-  ( y  =  z  ->  ( [. y  /  x ]. ph  <->  [. z  /  x ]. ph ) )
3 dfsbcq 2968 . 2  |-  ( y  =  suc  z  -> 
( [. y  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
4 sbceq2a 2977 . 2  |-  ( y  =  x  ->  ( [. y  /  x ]. ph  <->  ph ) )
5 tfindes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1629 . . . 4  |-  F/ x  z  e.  On
7 nfsbc1v 2985 . . . . 5  |-  F/ x [. z  /  x ]. ph
8 nfsbc1v 2985 . . . . 5  |-  F/ x [. suc  z  /  x ]. ph
97, 8nfim 1735 . . . 4  |-  F/ x
( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
106, 9nfim 1735 . . 3  |-  F/ x
( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
11 eleq1 2318 . . . 4  |-  ( x  =  z  ->  (
x  e.  On  <->  z  e.  On ) )
12 sbceq1a 2976 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
13 suceq 4429 . . . . . 6  |-  ( x  =  z  ->  suc  x  =  suc  z )
14 dfsbcq 2968 . . . . . 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 1879 . 2  |-  ( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
20 cbvralsv 2750 . . . 4  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [
z  /  x ] ph )
21 sbsbc 2970 . . . . 5  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
2221ralbii 2542 . . . 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 4622 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   [wsb 1883   A.wral 2518   [.wsbc 2966   (/)c0 3430   Oncon0 4364   Lim wlim 4365   suc csuc 4366
This theorem is referenced by:  tfinds2  4626
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370
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