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Theorem findes 4381
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 [∅ / 𝑥]𝜑
findes.2 (𝑥 ∈ ω → (𝜑[suc 𝑥 / 𝑥]𝜑))
Assertion
Ref Expression
findes (𝑥 ∈ ω → 𝜑)

Proof of Theorem findes
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2829 . 2 (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 sbequ 1763 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3 dfsbcq2 2829 . 2 (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
4 sbequ12r 1697 . 2 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
5 findes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1462 . . . 4 𝑥 𝑦 ∈ ω
7 nfs1v 1858 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
8 nfsbc1v 2844 . . . . 5 𝑥[suc 𝑦 / 𝑥]𝜑
97, 8nfim 1505 . . . 4 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
106, 9nfim 1505 . . 3 𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
11 eleq1 2145 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
12 sbequ12 1696 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13 suceq 4193 . . . . . 6 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
14 dfsbcq 2828 . . . . . 6 (suc 𝑥 = suc 𝑦 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
1513, 14syl 14 . . . . 5 (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
1612, 15imbi12d 232 . . . 4 (𝑥 = 𝑦 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)))
1711, 16imbi12d 232 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ ω → (𝜑[suc 𝑥 / 𝑥]𝜑)) ↔ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))))
18 findes.2 . . 3 (𝑥 ∈ ω → (𝜑[suc 𝑥 / 𝑥]𝜑))
1910, 17, 18chvar 1682 . 2 (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
201, 2, 3, 4, 5, 19finds 4378 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  [wsb 1687  [wsbc 2826  c0 3269  suc csuc 4156  ωcom 4368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-suc 4162  df-iom 4369
This theorem is referenced by: (None)
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