Theorem findes 3983

Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 3946 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)

Hypotheses

Ref	Expression
findes.1	|- [(/) / x]ph
findes.2	|- (x e. om -> (ph -> [suc x / x]ph))

Assertion

Ref	Expression
findes	|- (x e. om -> ph)

Proof of Theorem findes

Step	Hyp	Ref	Expression
1		dfsbcq 2455	. 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2		sbequ 1599	. 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3		dfsbcq 2455	. 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4		sbequ12r 1546	. 2 |- (z = x -> ([z / x]ph <-> ph))
5		findes.1	. 2 |- [(/) / x]ph
6		ax-17 1317	. . . 4 |- (y e. om -> A.x y e. om)
7		hbs1 1722	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8		visset 2295	. . . . . . 7 |- y e. _V
9	8	sucex 3892	. . . . . 6 |- suc y e. _V
10	9	hbsbc1v 2464	. . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
11	7, 10	hbim 1354	. . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
12	6, 11	hbim 1354	. . 3 |- ((y e. om -> ([y / x]ph -> [suc y / x]ph)) -> A.x(y e. om -> ([y / x]ph -> [suc y / x]ph)))
13		eleq1 1957	. . . 4 |- (x = y -> (x e. om <-> y e. om))
14		sbequ12 1545	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15		suceq 3729	. . . . . 6 |- (x = y -> suc x = suc y)
16		dfsbcq 2455	. . . . . 6 |- (suc x = suc y -> ([suc x / x]ph <-> [suc y / x]ph))
17	15, 16	syl 12	. . . . 5 |- (x = y -> ([suc x / x]ph <-> [suc y / x]ph))
18	14, 17	imbi12d 688	. . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
19	13, 18	imbi12d 688	. . 3 |- (x = y -> ((x e. om -> (ph -> [suc x / x]ph)) <-> (y e. om -> ([y / x]ph -> [suc y / x]ph))))
20		findes.2	. . 3 |- (x e. om -> (ph -> [suc x / x]ph))
21	12, 19, 20	chvar 1530	. 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
22	1, 2, 3, 4, 5, 21	finds 3979	1 |- (x e. om -> ph)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  [wsbc 1534  (/)c0 2875  suc csuc 3659  omcom 3949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950

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