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Theorem findes 4875
 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 4842 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)
Hypotheses
Ref Expression
findes.1
findes.2
Assertion
Ref Expression
findes

Proof of Theorem findes
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3164 . 2
2 sbequ 2111 . 2
3 dfsbcq2 3164 . 2
4 sbequ12r 1945 . 2
5 findes.1 . 2
6 nfv 1629 . . . 4
7 nfs1v 2182 . . . . 5
8 nfsbc1v 3180 . . . . 5
97, 8nfim 1832 . . . 4
106, 9nfim 1832 . . 3
11 eleq1 2496 . . . 4
12 sbequ12 1944 . . . . 5
13 suceq 4646 . . . . . 6
14 dfsbcq 3163 . . . . . 6
1513, 14syl 16 . . . . 5
1612, 15imbi12d 312 . . . 4
1711, 16imbi12d 312 . . 3
18 findes.2 . . 3
1910, 17, 18chvar 1968 . 2
201, 2, 3, 4, 5, 19finds 4871 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652  wsb 1658   wcel 1725  wsbc 3161  c0 3628   csuc 4583  com 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846
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