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Theorem findes 6800
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 6766 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 3294 . 2 (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑[∅ / 𝑥]𝜑))
2 sbequ 2259 . 2 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3 dfsbcq2 3294 . 2 (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
4 sbequ12r 2132 . 2 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
5 findes.1 . 2 [∅ / 𝑥]𝜑
6 nfv 1792 . . . 4 𝑥 𝑦 ∈ ω
7 nfs1v 2320 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
8 nfsbc1v 3311 . . . . 5 𝑥[suc 𝑦 / 𝑥]𝜑
97, 8nfim 2056 . . . 4 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
106, 9nfim 2056 . . 3 𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
11 eleq1 2571 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
12 sbequ12 2131 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13 suceq 5539 . . . . . 6 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1413sbceq1d 3296 . . . . 5 (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
1512, 14imbi12d 329 . . . 4 (𝑥 = 𝑦 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)))
1611, 15imbi12d 329 . . 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 2153 . 2 (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
191, 2, 3, 4, 5, 18finds 6796 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 1828  wcel 1937  [wsbc 3291  c0 3757  suc csuc 5476  ωcom 6769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pr 4680  ax-un 6659
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3068  df-sbc 3292  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-br 4435  df-opab 4494  df-tr 4531  df-eprel 4791  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-om 6770
This theorem is referenced by:  rdgeqoa  31994
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