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Theorem finds1 7242
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 2771 . 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 7241 . 2 (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
101, 9mpi 20 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  c0 4063  suc csuc 5868  ωcom 7212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-om 7213
This theorem is referenced by:  findcard  8355  findcard2  8356  pwfi  8417  alephfplem3  9129  pwsdompw  9228  hsmexlem4  9453
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