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Theorem tfindes 4669
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 3006 . 2  |-  ( y  =  (/)  ->  ( [. y  /  x ]. ph  <->  [. (/)  /  x ]. ph ) )
2 dfsbcq 3006 . 2  |-  ( y  =  z  ->  ( [. y  /  x ]. ph  <->  [. z  /  x ]. ph ) )
3 dfsbcq 3006 . 2  |-  ( y  =  suc  z  -> 
( [. y  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
4 sbceq2a 3015 . 2  |-  ( y  =  x  ->  ( [. y  /  x ]. ph  <->  ph ) )
5 tfindes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1609 . . . 4  |-  F/ x  z  e.  On
7 nfsbc1v 3023 . . . . 5  |-  F/ x [. z  /  x ]. ph
8 nfsbc1v 3023 . . . . 5  |-  F/ x [. suc  z  /  x ]. ph
97, 8nfim 1781 . . . 4  |-  F/ x
( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
106, 9nfim 1781 . . 3  |-  F/ x
( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
11 eleq1 2356 . . . 4  |-  ( x  =  z  ->  (
x  e.  On  <->  z  e.  On ) )
12 sbceq1a 3014 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
13 suceq 4473 . . . . . 6  |-  ( x  =  z  ->  suc  x  =  suc  z )
14 dfsbcq 3006 . . . . . 6  |-  ( suc  x  =  suc  z  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  z  /  x ]. ph )
)
1513, 14syl 15 . . . . 5  |-  ( x  =  z  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
1612, 15imbi12d 311 . . . 4  |-  ( x  =  z  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  (
[. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
) )
1711, 16imbi12d 311 . . 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 1939 . 2  |-  ( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
20 cbvralsv 2788 . . . 4  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [
z  /  x ] ph )
21 sbsbc 3008 . . . . 5  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
2221ralbii 2580 . . . 4  |-  ( A. z  e.  y  [
z  /  x ] ph 
<-> 
A. z  e.  y 
[. z  /  x ]. ph )
2320, 22bitri 240 . . 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 209 . 2  |-  ( Lim  y  ->  ( A. z  e.  y  [. z  /  x ]. ph  ->  [. y  /  x ]. ph ) )
261, 2, 3, 4, 5, 19, 25tfinds 4666 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632   [wsb 1638    e. wcel 1696   A.wral 2556   [.wsbc 3004   (/)c0 3468   Oncon0 4408   Lim wlim 4409   suc csuc 4410
This theorem is referenced by:  tfinds2  4670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414
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