Theorem finds2 7921

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 = ∅ → (𝜑 ↔ 𝜓))
finds2.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
finds2.3	⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
finds2.4	⊢ (𝜏 → 𝜓)
finds2.5	⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃)))

Assertion

Ref	Expression
finds2	⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))

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

Step	Hyp	Ref	Expression
1		finds2.4	. . . . 5 ⊢ (𝜏 → 𝜓)
2		0ex 5313	. . . . . 6 ⊢ ∅ ∈ V
3		finds2.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 340	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	elab 3681	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜓))
6	1, 5	mpbir 231	. . . 4 ⊢ ∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)}
7		finds2.5	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃)))
8	7	a2d 29	. . . . . 6 ⊢ (𝑦 ∈ ω → ((𝜏 → 𝜒) → (𝜏 → 𝜃)))
9		vex 3482	. . . . . . 7 ⊢ 𝑦 ∈ V
10		finds2.2	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
11	10	imbi2d 340	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒)))
12	9, 11	elab 3681	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒))
13	9	sucex 7826	. . . . . . 7 ⊢ suc 𝑦 ∈ V
14		finds2.3	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
15	14	imbi2d 340	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃)))
16	13, 15	elab 3681	. . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜃))
17	8, 12, 16	3imtr4g 296	. . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)}))
18	17	rgen 3061	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
19		peano5 7916	. . . 4 ⊢ ((∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)})
20	6, 18, 19	mp2an 692	. . 3 ⊢ ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)}
21	20	sseli 3991	. 2 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
22		abid 2716	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜑))
23	21, 22	sylib 218	1 ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059   ⊆ wss 3963  ∅c0 4339  suc csuc 6388  ωcom 7887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888

This theorem is referenced by:  finds1  7922  onnseq  8383  nnacl  8648  nnmcl  8649  nnecl  8650  nnacom  8654  nnaass  8659  nndi  8660  nnmass  8661  nnmsucr  8662  nnmcom  8663  nnmordi  8668  omsmolem  8694  isinf  9294  isinfOLD  9295  unblem2  9327  fiint  9364  fiintOLD  9365  dffi3  9469  card2inf  9593  cantnfle  9709  cantnflt  9710  cantnflem1  9727  cnfcom  9738  trcl  9766  fseqenlem1  10062  nnadju  10236  infpssrlem3  10343  fin23lem26  10363  axdc3lem2  10489  axdc4lem  10493  axdclem2  10558  wunr1om  10757  wuncval2  10785  tskr1om  10805  grothomex  10867  peano5nni  12267  precsexlem6  28251  precsexlem7  28252  noseqind  28313  om2noseqlt  28320  neibastop2lem  36343  finxpreclem6  37379  domalom  37387  oaabsb  43284