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Theorem findes 4326
 Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (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 2767 . 2
2 sbequ 1721 . 2
3 dfsbcq2 2767 . 2
4 sbequ12r 1655 . 2
5 findes.1 . 2
6 nfv 1421 . . . 4
7 nfs1v 1815 . . . . 5
8 nfsbc1v 2782 . . . . 5
97, 8nfim 1464 . . . 4
106, 9nfim 1464 . . 3
11 eleq1 2100 . . . 4
12 sbequ12 1654 . . . . 5
13 suceq 4139 . . . . . 6
14 dfsbcq 2766 . . . . . 6
1513, 14syl 14 . . . . 5
1612, 15imbi12d 223 . . . 4
1711, 16imbi12d 223 . . 3
18 findes.2 . . 3
1910, 17, 18chvar 1640 . 2
201, 2, 3, 4, 5, 19finds 4323 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98   wceq 1243   wcel 1393  wsb 1645  wsbc 2764  c0 3224   csuc 4102  com 4313 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314 This theorem is referenced by: (None)
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