Theorem findes 4148

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 4114 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 2732	. 2 |- (z = (/) -> ([z / x]ph <-> [(/) / x]ph))
2		sbequ 1904	. 2 |- (z = y -> ([z / x]ph <-> [y / x]ph))
3		dfsbcq 2732	. 2 |- (z = suc y -> ([z / x]ph <-> [suc y / x]ph))
4		sbequ12r 1855	. 2 |- (z = x -> ([z / x]ph <-> ph))
5		findes.1	. 2 |- [(/) / x]ph
6		ax-17 1634	. . . 4 |- (y e. om -> A.x y e. om)
7		hbs1 2014	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
8		visset 2572	. . . . . . 7 |- y e. _V
9	8	sucex 4067	. . . . . 6 |- suc y e. _V
10	9	hbsbc1v 2739	. . . . 5 |- ([suc y / x]ph -> A.x[suc y / x]ph)
11	7, 10	hbim 1672	. . . 4 |- (([y / x]ph -> [suc y / x]ph) -> A.x([y / x]ph -> [suc y / x]ph))
12	6, 11	hbim 1672	. . 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 2233	. . . 4 |- (x = y -> (x e. om <-> y e. om))
14		sbequ12 1854	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
15		suceq 3907	. . . . . 6 |- (x = y -> suc x = suc y)
16		dfsbcq 2732	. . . . . 6 |- (suc x = suc y -> ([suc x / x]ph <-> [suc y / x]ph))
17	15, 16	syl 13	. . . . 5 |- (x = y -> ([suc x / x]ph <-> [suc y / x]ph))
18	14, 17	imbi12d 384	. . . 4 |- (x = y -> ((ph -> [suc x / x]ph) <-> ([y / x]ph -> [suc y / x]ph)))
19	13, 18	imbi12d 384	. . 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 1839	. 2 |- (y e. om -> ([y / x]ph -> [suc y / x]ph))
22	1, 2, 3, 4, 5, 21	finds 4144	1 |- (x e. om -> ph)

Syntax hints:   -> wi 3   <-> wb 231   = wceq 1615   e. wcel 1617  [wsbc 1843  (/)c0 3114  suc csuc 3845  omcom 4117

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-10 1625  ax-11 1626  ax-12 1627  ax-13 1628  ax-14 1629  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152  ax-sep 3638  ax-nul 3645  ax-pow 3681  ax-pr 3719  ax-un 3961

This theorem depends on definitions:  df-bi 232  df-or 434  df-an 435  df-3or 1131  df-3an 1132  df-ex 1645  df-sb 1845  df-eu 2070  df-mo 2071  df-clab 2158  df-cleq 2163  df-clel 2166  df-ne 2297  df-ral 2389  df-rex 2390  df-rab 2392  df-v 2571  df-sbc 2731  df-dif 2862  df-un 2864  df-in 2866  df-ss 2868  df-pss 2870  df-nul 3115  df-if 3213  df-pw 3261  df-sn 3274  df-pr 3275  df-tp 3277  df-op 3278  df-uni 3399  df-br 3540  df-opab 3598  df-tr 3612  df-eprel 3776  df-po 3784  df-so 3796  df-fr 3814  df-we 3830  df-ord 3846  df-on 3847  df-lim 3848  df-suc 3849  df-om 4118

Copyright terms: Public domain