Theorem finds2 7879

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 5257	. . . . . 6 ⊢ ∅ ∈ V
3		finds2.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 342	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	elab 3638	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜓))
6	1, 5	mpbir 233	. . . 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 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
10		finds2.2	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
11	10	imbi2d 342	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒)))
12	9, 11	elab 3638	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒))
13	9	sucex 7789	. . . . . . 7 ⊢ suc 𝑦 ∈ V
14		finds2.3	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
15	14	imbi2d 342	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃)))
16	13, 15	elab 3638	. . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜃))
17	8, 12, 16	3imtr4g 298	. . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)}))
18	17	rgen 3078	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
19		peano5 7874	. . . 4 ⊢ ((∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)})
20	6, 18, 19	mp2an 702	. . 3 ⊢ ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)}
21	20	sseli 3932	. 2 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
22		abid 2744	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜑))
23	21, 22	sylib 220	1 ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560   ∈ wcel 2142  {cab 2740  ∀wral 3076   ⊆ wss 3904  ∅c0 4285  suc csuc 6348  ωcom 7846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-om 7847

This theorem is referenced by:  finds1  7880  onnseq  8315  nnacl  8581  nnmcl  8582  nnecl  8583  nnacom  8587  nnaass  8592  nndi  8593  nnmass  8594  nnmsucr  8595  nnmcom  8596  nnmordi  8601  omsmolem  8627  isinf  9209  unblem2  9237  fiint  9271  dffi3  9377  card2inf  9503  cantnfle  9626  cantnflt  9627  cantnflem1  9644  cnfcom  9655  trcl  9683  fseqenlem1  9980  nnadju  10154  infpssrlem3  10262  fin23lem26  10282  axdc3lem2  10408  axdc4lem  10412  axdclem2  10477  wunr1om  10677  wuncval2  10705  tskr1om  10725  grothomex  10787  peano5nni  12213  precsexlem6  28302  precsexlem7  28303  noseqind  28382  om2noseqlt  28389  fineqvinfep  35418  neibastop2lem  36717  ttcmin  36853  dfttc2g  36863  mh-inf3f1  36898  finxpreclem6  37887  domalom  37895  oaabsb  43868