Theorem finds2 7838

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 5250	. . . . . 6 ⊢ ∅ ∈ V
3		finds2.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 340	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	elab 3632	. . . . 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 3442	. . . . . . 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 3632	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒))
13	9	sucex 7749	. . . . . . 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 3632	. . . . . 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 3051	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
19		peano5 7833	. . . 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 3927	. 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 1541   ∈ wcel 2113  {cab 2712  ∀wral 3049   ⊆ wss 3899  ∅c0 4283  suc csuc 6317  ωcom 7806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-om 7807

This theorem is referenced by:  finds1  7839  onnseq  8274  nnacl  8537  nnmcl  8538  nnecl  8539  nnacom  8543  nnaass  8548  nndi  8549  nnmass  8550  nnmsucr  8551  nnmcom  8552  nnmordi  8557  omsmolem  8583  isinf  9163  unblem2  9191  fiint  9225  dffi3  9332  card2inf  9458  cantnfle  9578  cantnflt  9579  cantnflem1  9596  cnfcom  9607  trcl  9635  fseqenlem1  9932  nnadju  10106  infpssrlem3  10213  fin23lem26  10233  axdc3lem2  10359  axdc4lem  10363  axdclem2  10428  wunr1om  10628  wuncval2  10656  tskr1om  10676  grothomex  10738  peano5nni  12146  precsexlem6  28180  precsexlem7  28181  noseqind  28253  om2noseqlt  28260  fineqvinfep  35230  neibastop2lem  36503  finxpreclem6  37540  domalom  37548  oaabsb  43478