Theorem findes 3247

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 3214 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 1987	. 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2		sbequ 1265	. 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3		dfsbcq 1987	. 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4		sbequ12r 1218	. 2 |- (z = x -> ([z / x]ph <-> ph))
5		findes.1	. 2 |- [(/) / x]ph
6		ax-17 1006	. . . 4 |- (y e. om -> A.x y e. om)
7		hbs1 1370	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8		visset 1858	. . . . . . 7 |- y e. V
9	8	sucex 3167	. . . . . 6 |- suc y e. V
10	9	hbsbc1v 1994	. . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
11	7, 10	hbim 1042	. . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
12	6, 11	hbim 1042	. . 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 1576	. . . 4 |- (x = y -> (x e. om <-> y e. om))
14		sbequ12 1217	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15		suceq 3037	. . . . . 6 |- (x = y -> suc x = suc y)
16		dfsbcq 1987	. . . . . 6 |- (suc x = suc y -> ([suc x / x]ph <-> [suc y / x]ph))
17	15, 16	syl 10	. . . . 5 |- (x = y -> ([suc x / x]ph <-> [suc y / x]ph))
18	14, 17	imbi12d 628	. . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
19	13, 18	imbi12d 628	. . 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 1203	. 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
22	1, 2, 3, 4, 5, 21	finds 3243	1 |- (x e. om -> ph)

Syntax hints:   -> wi 3   <-> wb 144   = wceq 991   e. wcel 993  [wsbc 1206  (/)c0 2331  suc csuc 2976  omcom 3217

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 997  ax-gen 998  ax-8 999  ax-9 1000  ax-10 1001  ax-11 1002  ax-12 1003  ax-13 1004  ax-14 1005  ax-17 1006  ax-4 1008  ax-5o 1010  ax-6o 1013  ax-9o 1158  ax-10o 1176  ax-16 1246  ax-11o 1254  ax-ext 1499  ax-sep 2776  ax-nul 2783  ax-pow 2817  ax-pr 2854  ax-un 3088

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 781  df-3an 782  df-ex 1016  df-sb 1208  df-eu 1420  df-mo 1421  df-clab 1505  df-cleq 1510  df-clel 1513  df-ne 1629  df-ral 1694  df-rex 1695  df-v 1857  df-sbc 1986  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2415  df-pw 2458  df-sn 2469  df-pr 2470  df-tp 2472  df-op 2473  df-uni 2569  df-br 2692  df-opab 2740  df-tr 2754  df-eprel 2909  df-po 2917  df-so 2928  df-fr 2946  df-we 2961  df-ord 2977  df-on 2978  df-lim 2979  df-suc 2980  df-om 3218

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