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Theorem finds1 7042
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
Hypotheses
Ref Expression
finds1.1 (𝑥 = ∅ → (𝜑𝜓))
finds1.2 (𝑥 = 𝑦 → (𝜑𝜒))
finds1.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
finds1.4 𝜓
finds1.5 (𝑦 ∈ ω → (𝜒𝜃))
Assertion
Ref Expression
finds1 (𝑥 ∈ ω → 𝜑)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)

Proof of Theorem finds1
StepHypRef Expression
1 eqid 2621 . 2 ∅ = ∅
2 finds1.1 . . 3 (𝑥 = ∅ → (𝜑𝜓))
3 finds1.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜒))
4 finds1.3 . . 3 (𝑥 = suc 𝑦 → (𝜑𝜃))
5 finds1.4 . . . 4 𝜓
65a1i 11 . . 3 (∅ = ∅ → 𝜓)
7 finds1.5 . . . 4 (𝑦 ∈ ω → (𝜒𝜃))
87a1d 25 . . 3 (𝑦 ∈ ω → (∅ = ∅ → (𝜒𝜃)))
92, 3, 4, 6, 8finds2 7041 . 2 (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
101, 9mpi 20 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  c0 3891  suc csuc 5684  ωcom 7012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-om 7013
This theorem is referenced by:  findcard  8143  findcard2  8144  pwfi  8205  alephfplem3  8873  pwsdompw  8970  hsmexlem4  9195
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