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Theorem finds2 4599
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, 29-Nov-2002.)
Hypotheses
Ref Expression
finds2.1 (𝑥 = ∅ → (𝜑𝜓))
finds2.2 (𝑥 = 𝑦 → (𝜑𝜒))
finds2.3 (𝑥 = suc 𝑦 → (𝜑𝜃))
finds2.4 (𝜏𝜓)
finds2.5 (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))
Assertion
Ref Expression
finds2 (𝑥 ∈ ω → (𝜏𝜑))
Distinct variable groups:   𝑥,𝑦,𝜏   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)

Proof of Theorem finds2
StepHypRef Expression
1 finds2.4 . . . . 5 (𝜏𝜓)
2 0ex 4129 . . . . . 6 ∅ ∈ V
3 finds2.1 . . . . . . 7 (𝑥 = ∅ → (𝜑𝜓))
43imbi2d 230 . . . . . 6 (𝑥 = ∅ → ((𝜏𝜑) ↔ (𝜏𝜓)))
52, 4elab 2881 . . . . 5 (∅ ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜓))
61, 5mpbir 146 . . . 4 ∅ ∈ {𝑥 ∣ (𝜏𝜑)}
7 finds2.5 . . . . . . 7 (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))
87a2d 26 . . . . . 6 (𝑦 ∈ ω → ((𝜏𝜒) → (𝜏𝜃)))
9 vex 2740 . . . . . . 7 𝑦 ∈ V
10 finds2.2 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜒))
1110imbi2d 230 . . . . . . 7 (𝑥 = 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜒)))
129, 11elab 2881 . . . . . 6 (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜒))
139sucex 4497 . . . . . . 7 suc 𝑦 ∈ V
14 finds2.3 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝜑𝜃))
1514imbi2d 230 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝜏𝜑) ↔ (𝜏𝜃)))
1613, 15elab 2881 . . . . . 6 (suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜃))
178, 12, 163imtr4g 205 . . . . 5 (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)}))
1817rgen 2530 . . . 4 𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)})
19 peano5 4596 . . . 4 ((∅ ∈ {𝑥 ∣ (𝜏𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏𝜑)})
206, 18, 19mp2an 426 . . 3 ω ⊆ {𝑥 ∣ (𝜏𝜑)}
2120sseli 3151 . 2 (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏𝜑)})
22 abid 2165 . 2 (𝑥 ∈ {𝑥 ∣ (𝜏𝜑)} ↔ (𝜏𝜑))
2321, 22sylib 122 1 (𝑥 ∈ ω → (𝜏𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wss 3129  c0 3422  suc csuc 4364  ωcom 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-iinf 4586
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4370  df-iom 4589
This theorem is referenced by:  finds1  4600  frecrdg  6406  nnacl  6478  nnmcl  6479  nnacom  6482  nnaass  6483  nndi  6484  nnmass  6485  nnmsucr  6486  nnmcom  6487  nnsucsssuc  6490  nntri3or  6491  nnaordi  6506  nnaword  6509  nnmordi  6514  nnaordex  6526  fiintim  6925  prarloclem3  7493  frec2uzuzd  10397  frec2uzrdg  10404
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