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Theorem findes 7889
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. See tfindes 7848 for the transfinite version. 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 3775 . 2 (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 sbequ 2078 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3 dfsbcq2 3775 . 2 (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
4 sbequ12r 2236 . 2 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
5 findes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1909 . . . 4 𝑥 𝑦 ∈ ω
7 nfs1v 2145 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
8 nfsbc1v 3792 . . . . 5 𝑥[suc 𝑦 / 𝑥]𝜑
97, 8nfim 1891 . . . 4 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
106, 9nfim 1891 . . 3 𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
11 eleq1w 2810 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
12 sbequ12 2235 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13 suceq 6423 . . . . . 6 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1413sbceq1d 3777 . . . . 5 (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
1512, 14imbi12d 344 . . . 4 (𝑥 = 𝑦 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)))
1611, 15imbi12d 344 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ ω → (𝜑[suc 𝑥 / 𝑥]𝜑)) ↔ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))))
17 findes.2 . . 3 (𝑥 ∈ ω → (𝜑[suc 𝑥 / 𝑥]𝜑))
1810, 16, 17chvarfv 2225 . 2 (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
191, 2, 3, 4, 5, 18finds 7885 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2059  wcel 2098  [wsbc 3772  c0 4317  suc csuc 6359  ωcom 7851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-om 7852
This theorem is referenced by:  rdgeqoa  36758
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