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Theorem findes 7253
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 7219 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 3571 . 2 (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 sbequ 2505 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3 dfsbcq2 3571 . 2 (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
4 sbequ12r 2251 . 2 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
5 findes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1984 . . . 4 𝑥 𝑦 ∈ ω
7 nfs1v 2566 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
8 nfsbc1v 3588 . . . . 5 𝑥[suc 𝑦 / 𝑥]𝜑
97, 8nfim 1966 . . . 4 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
106, 9nfim 1966 . . 3 𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
11 eleq1w 2814 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
12 sbequ12 2250 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13 suceq 5943 . . . . . 6 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1413sbceq1d 3573 . . . . 5 (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
1512, 14imbi12d 333 . . . 4 (𝑥 = 𝑦 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)))
1611, 15imbi12d 333 . . 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, 17chvar 2399 . 2 (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
191, 2, 3, 4, 5, 18finds 7249 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2038  wcel 2131  [wsbc 3568  c0 4050  suc csuc 5878  ωcom 7222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-om 7223
This theorem is referenced by:  rdgeqoa  33521  cnfinltrel  33544
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