Theorem tfindes 7062

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.

Step	Hyp	Ref	Expression
1		dfsbcq 3437	. 2 |- ( y = (/) -> ( [. y / x ]. ph <-> [. (/) / x ]. ph ) )
2		dfsbcq 3437	. 2 |- ( y = z -> ( [. y / x ]. ph <-> [. z / x ]. ph ) )
3		dfsbcq 3437	. 2 |- ( y = suc z -> ( [. y / x ]. ph <-> [. suc z / x ]. ph ) )
4		sbceq2a 3447	. 2 |- ( y = x -> ( [. y / x ]. ph <-> ph ) )
5		tfindes.1	. 2 |- [. (/) / x ]. ph
6		nfv 1843	. . . 4 |- F/ x z e. On
7		nfsbc1v 3455	. . . . 5 |- F/ x [. z / x ]. ph
8		nfsbc1v 3455	. . . . 5 |- F/ x [. suc z / x ]. ph
9	7, 8	nfim 1825	. . . 4 |- F/ x ( [. z / x ]. ph -> [. suc z / x ]. ph )
10	6, 9	nfim 1825	. . 3 |- F/ x ( z e. On -> ( [. z / x ]. ph -> [. suc z / x ]. ph ) )
11		eleq1 2689	. . . 4 |- ( x = z -> ( x e. On <-> z e. On ) )
12		sbceq1a 3446	. . . . 5 |- ( x = z -> ( ph <-> [. z / x ]. ph ) )
13		suceq 5790	. . . . . 6 |- ( x = z -> suc x = suc z )
14	13	sbceq1d 3440	. . . . 5 |- ( x = z -> ( [. suc x / x ]. ph <-> [. suc z / x ]. ph ) )
15	12, 14	imbi12d 334	. . . 4 |- ( x = z -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [. z / x ]. ph -> [. suc z / x ]. ph ) ) )
16	11, 15	imbi12d 334	. . 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 ) )
18	10, 16, 17	chvar 2262	. 2 |- ( z e. On -> ( [. z / x ]. ph -> [. suc z / x ]. ph ) )
19		cbvralsv 3182	. . . 4 |- ( A. x e. y ph <-> A. z e. y [ z / x ] ph )
20		sbsbc 3439	. . . . 5 |- ( [ z / x ] ph <-> [. z / x ]. ph )
21	20	ralbii 2980	. . . 4 |- ( A. z e. y [ z / x ] ph <-> A. z e. y [. z / x ]. ph )
22	19, 21	bitri 264	. . 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 ) )
24	22, 23	syl5bir 233	. 2 |- ( Lim y -> ( A. z e. y [. z / x ]. ph -> [. y / x ]. ph ) )
25	1, 2, 3, 4, 5, 18, 24	tfinds 7059	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   [wsb 1880   e. wcel 1990   A.wral 2912   [.wsbc 3435   (/)c0 3915   Oncon0 5723   Lim wlim 5724   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729

This theorem is referenced by:  tfinds2  7063