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Theorem findes 7900
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 7859 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 3777 . 2 (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 sbequ 2079 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3 dfsbcq2 3777 . 2 (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
4 sbequ12r 2237 . 2 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
5 findes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1910 . . . 4 𝑥 𝑦 ∈ ω
7 nfs1v 2146 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
8 nfsbc1v 3794 . . . . 5 𝑥[suc 𝑦 / 𝑥]𝜑
97, 8nfim 1892 . . . 4 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
106, 9nfim 1892 . . 3 𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
11 eleq1w 2811 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
12 sbequ12 2236 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13 suceq 6429 . . . . . 6 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1413sbceq1d 3779 . . . . 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 2226 . 2 (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
191, 2, 3, 4, 5, 18finds 7896 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2060  wcel 2099  [wsbc 3774  c0 4318  suc csuc 6365  ωcom 7862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-om 7863
This theorem is referenced by:  rdgeqoa  36772
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