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Theorem finds1 7776
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 2736 . 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 7775 . 2 (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
101, 9mpi 20 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2104  c0 4262  suc csuc 6279  ωcom 7740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7616
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-tr 5199  df-eprel 5502  df-po 5510  df-so 5511  df-fr 5551  df-we 5553  df-ord 6280  df-on 6281  df-lim 6282  df-suc 6283  df-om 7741
This theorem is referenced by:  findcard  8980  findcard2  8981  findcard2OLD  9096  pwfiOLD  9154  alephfplem3  9904  pwsdompw  10002  hsmexlem4  10227
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