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Theorem tfindes 6696
Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction step for successors, and the third is the induction step 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 3329 . 2  |-  ( y  =  (/)  ->  ( [. y  /  x ]. ph  <->  [. (/)  /  x ]. ph ) )
2 dfsbcq 3329 . 2  |-  ( y  =  z  ->  ( [. y  /  x ]. ph  <->  [. z  /  x ]. ph ) )
3 dfsbcq 3329 . 2  |-  ( y  =  suc  z  -> 
( [. y  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
4 sbceq2a 3339 . 2  |-  ( y  =  x  ->  ( [. y  /  x ]. ph  <->  ph ) )
5 tfindes.1 . 2  |-  [. (/)  /  x ]. ph
6 nfv 1708 . . . 4  |-  F/ x  z  e.  On
7 nfsbc1v 3347 . . . . 5  |-  F/ x [. z  /  x ]. ph
8 nfsbc1v 3347 . . . . 5  |-  F/ x [. suc  z  /  x ]. ph
97, 8nfim 1921 . . . 4  |-  F/ x
( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
106, 9nfim 1921 . . 3  |-  F/ x
( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
11 eleq1 2529 . . . 4  |-  ( x  =  z  ->  (
x  e.  On  <->  z  e.  On ) )
12 sbceq1a 3338 . . . . 5  |-  ( x  =  z  ->  ( ph 
<-> 
[. z  /  x ]. ph ) )
13 suceq 4952 . . . . . 6  |-  ( x  =  z  ->  suc  x  =  suc  z )
1413sbceq1d 3332 . . . . 5  |-  ( x  =  z  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  z  /  x ]. ph ) )
1512, 14imbi12d 320 . . . 4  |-  ( x  =  z  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  (
[. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
) )
1611, 15imbi12d 320 . . 3  |-  ( x  =  z  ->  (
( x  e.  On  ->  ( ph  ->  [. suc  x  /  x ]. ph )
)  <->  ( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [.
suc  z  /  x ]. ph ) ) ) )
17 tfindes.2 . . 3  |-  ( x  e.  On  ->  ( ph  ->  [. suc  x  /  x ]. ph ) )
1810, 16, 17chvar 2014 . 2  |-  ( z  e.  On  ->  ( [. z  /  x ]. ph  ->  [. suc  z  /  x ]. ph )
)
19 cbvralsv 3095 . . . 4  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [
z  /  x ] ph )
20 sbsbc 3331 . . . . 5  |-  ( [ z  /  x ] ph 
<-> 
[. z  /  x ]. ph )
2120ralbii 2888 . . . 4  |-  ( A. z  e.  y  [
z  /  x ] ph 
<-> 
A. z  e.  y 
[. z  /  x ]. ph )
2219, 21bitri 249 . . 3  |-  ( A. x  e.  y  ph  <->  A. z  e.  y  [. z  /  x ]. ph )
23 tfindes.3 . . 3  |-  ( Lim  y  ->  ( A. x  e.  y  ph  ->  [. y  /  x ]. ph ) )
2422, 23syl5bir 218 . 2  |-  ( Lim  y  ->  ( A. z  e.  y  [. z  /  x ]. ph  ->  [. y  /  x ]. ph ) )
251, 2, 3, 4, 5, 18, 24tfinds 6693 1  |-  ( x  e.  On  ->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   [wsb 1740    e. wcel 1819   A.wral 2807   [.wsbc 3327   (/)c0 3793   Oncon0 4887   Lim wlim 4888   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893
This theorem is referenced by:  tfinds2  6697
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