Theorem finds2 3158

Description: Principle of Finite Induction (inference schema) with 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 2711	. . . . . 6 |- (/) e. V
3		finds2.1	. . . . . . 7 |- (x = (/) -> (ph <-> ps))
4	3	imbi2d 612	. . . . . 6 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
5	2, 4	elab 1897	. . . . 5 |- ((/) e. {x | (ta -> ph)} <-> (ta -> ps))
6	1, 5	mpbir 190	. . . 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 1813	. . . . . . 7 |- y e. V
10		finds2.2	. . . . . . . 8 |- (x = y -> (ph <-> ch))
11	10	imbi2d 612	. . . . . . 7 |- (x = y -> ((ta -> ph) <-> (ta -> ch)))
12	9, 11	elab 1897	. . . . . 6 |- (y e. {x | (ta -> ph)} <-> (ta -> ch))
13	9	sucex 3050	. . . . . . 7 |- suc y e. V
14		finds2.3	. . . . . . . 8 |- (x = suc y -> (ph <-> th))
15	14	imbi2d 612	. . . . . . 7 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
16	13, 15	elab 1897	. . . . . 6 |- (suc y e. {x | (ta -> ph)} <-> (ta -> th))
17	8, 12, 16	3imtr4g 553	. . . . 5 |- (y e. om -> (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)}))
18	17	rgen 1698	. . . 4 |- A.y e. om (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)})
19		peano5 3153	. . . 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 697	. . 3 |- om (_ {x | (ta -> ph)}
21	20	sseli 2065	. 2 |- (x e. om -> x e. {x | (ta -> ph)})
22		abid 1465	. 2 |- (x e. {x | (ta -> ph)} <-> (ta -> ph))
23	21, 22	sylib 198	1 |- (x e. om -> (ta -> ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645   (_ wss 2047  (/)c0 2280  suc csuc 2950  omcom 3131

This theorem is referenced by:  finds1 3159  omsmolem 4256  unblem2 4541  fiint 4559  fiintOLD 4560  trcl 4645  alephfplem3 4898

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132

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