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Theorem finds1 4407
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 2088 . 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 9 . . 3 (∅ = ∅ → 𝜓)
7 finds1.5 . . . 4 (𝑦 ∈ ω → (𝜒𝜃))
87a1d 22 . . 3 (𝑦 ∈ ω → (∅ = ∅ → (𝜒𝜃)))
92, 3, 4, 6, 8finds2 4406 . 2 (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
101, 9mpi 15 1 (𝑥 ∈ ω → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wcel 1438  c0 3284  suc csuc 4183  ωcom 4395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-suc 4189  df-iom 4396
This theorem is referenced by:  findcard  6584  findcard2  6585  findcard2s  6586
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