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Theorem finds2 3148
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
StepHypRef Expression
1 finds2.4 . . . . 5 |- (ta -> ps)
2 0ex 2701 . . . . . 6 |- (/) e. V
3 finds2.1 . . . . . . 7 |- (x = (/) -> (ph <-> ps))
43imbi2d 610 . . . . . 6 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
52, 4elab 1888 . . . . 5 |- ((/) e. {x | (ta -> ph)} <-> (ta -> ps))
61, 5mpbir 190 . . . 4 |- (/) e. {x | (ta -> ph)}
7 finds2.5 . . . . . . 7 |- (y e. om -> (ta -> (ch -> th)))
87a2d 13 . . . . . 6 |- (y e. om -> ((ta -> ch) -> (ta -> th)))
9 visset 1804 . . . . . . 7 |- y e. V
10 finds2.2 . . . . . . . 8 |- (x = y -> (ph <-> ch))
1110imbi2d 610 . . . . . . 7 |- (x = y -> ((ta -> ph) <-> (ta -> ch)))
129, 11elab 1888 . . . . . 6 |- (y e. {x | (ta -> ph)} <-> (ta -> ch))
139sucex 3040 . . . . . . 7 |- suc y e. V
14 finds2.3 . . . . . . . 8 |- (x = suc y -> (ph <-> th))
1514imbi2d 610 . . . . . . 7 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
1613, 15elab 1888 . . . . . 6 |- (suc y e. {x | (ta -> ph)} <-> (ta -> th))
178, 12, 163imtr4g 551 . . . . 5 |- (y e. om -> (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)}))
1817rgen 1690 . . . 4 |- A.y e. om (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)})
19 peano5 3143 . . . 4 |- (((/) e. {x | (ta -> ph)} /\ A.y e. om (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)})) -> om (_ {x | (ta -> ph)})
206, 18, 19mp2an 695 . . 3 |- om (_ {x | (ta -> ph)}
2120sseli 2055 . 2 |- (x e. om -> x e. {x | (ta -> ph)})
22 abid 1458 . 2 |- (x e. {x | (ta -> ph)} <-> (ta -> ph))
2321, 22sylib 198 1 |- (x e. om -> (ta -> ph))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  {cab 1456  A.wral 1637   (_ wss 2037  (/)c0 2270  suc csuc 2940  omcom 3121
This theorem is referenced by:  finds1 3149  omsmolem 4240  unblem2 4518  fiint 4534  trcl 4617  alephfplem3 4870
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122
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