Theorem finds2 3246

Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136.

Hypotheses

Ref	Expression
finds2.1	|- (x = (/) -> (ph <-> ps))
finds2.2	|- (x = y -> (ph <-> ch))
finds2.3	|- (x = suc y -> (ph <-> th))
finds2.4	|- (ta -> ps)
finds2.5	|- (y e. om -> (ta -> (ch -> th)))

Assertion

Ref	Expression
finds2	|- (x e. om -> (ta -> ph))

Distinct variable groups:   x,y,ta   ps,x   ch,x   th,x   ph,y

Proof of Theorem finds2

Step	Hyp	Ref	Expression
1		finds2.4	. . . . 5 |- (ta -> ps)
2		0ex 2785	. . . . . 6 |- (/) e. V
3		finds2.1	. . . . . . 7 |- (x = (/) -> (ph <-> ps))
4	3	imbi2d 615	. . . . . 6 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
5	2, 4	elab 1943	. . . . 5 |- ((/) e. {x | (ta -> ph)} <-> (ta -> ps))
6	1, 5	mpbir 188	. . . 4 |- (/) e. {x | (ta -> ph)}
7		finds2.5	. . . . . . 7 |- (y e. om -> (ta -> (ch -> th)))
8	7	a2d 13	. . . . . 6 |- (y e. om -> ((ta -> ch) -> (ta -> th)))
9		visset 1859	. . . . . . 7 |- y e. V
10		finds2.2	. . . . . . . 8 |- (x = y -> (ph <-> ch))
11	10	imbi2d 615	. . . . . . 7 |- (x = y -> ((ta -> ph) <-> (ta -> ch)))
12	9, 11	elab 1943	. . . . . 6 |- (y e. {x | (ta -> ph)} <-> (ta -> ch))
13	9	sucex 3168	. . . . . . 7 |- suc y e. V
14		finds2.3	. . . . . . . 8 |- (x = suc y -> (ph <-> th))
15	14	imbi2d 615	. . . . . . 7 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
16	13, 15	elab 1943	. . . . . 6 |- (suc y e. {x | (ta -> ph)} <-> (ta -> th))
17	8, 12, 16	3imtr4g 556	. . . . 5 |- (y e. om -> (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)}))
18	17	rgen 1744	. . . 4 |- A.y e. om (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)})
19		peano5 3241	. . . 4 |- (((/) e. {x | (ta -> ph)} /\ A.y e. om (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)})) -> om (_ {x | (ta -> ph)})
20	6, 18, 19	mp2an 701	. . 3 |- om (_ {x | (ta -> ph)}
21	20	sseli 2117	. 2 |- (x e. om -> x e. {x | (ta -> ph)})
22		abid 1507	. 2 |- (x e. {x | (ta -> ph)} <-> (ta -> ph))
23	21, 22	sylib 196	1 |- (x e. om -> (ta -> ph))

Syntax hints:   -> wi 3   <-> wb 144   = wceq 992   e. wcel 994  {cab 1505  A.wral 1691   (_ wss 2099  (/)c0 2332  suc csuc 2977  omcom 3218

This theorem is referenced by:  finds1 3247  omsmolem 4396  unblem2 4687  fiint 4703  trcl 4791  alephfplem3 5048  nnacda 5090

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219

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