Theorem finds2 7260

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 4942	. . . . . 6 ⊢ ∅ ∈ V
3		finds2.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 329	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	elab 3490	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜓))
6	1, 5	mpbir 221	. . . 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 3343	. . . . . . 7 ⊢ 𝑦 ∈ V
10		finds2.2	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
11	10	imbi2d 329	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒)))
12	9, 11	elab 3490	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒))
13	9	sucex 7177	. . . . . . 7 ⊢ suc 𝑦 ∈ V
14		finds2.3	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
15	14	imbi2d 329	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃)))
16	13, 15	elab 3490	. . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜃))
17	8, 12, 16	3imtr4g 285	. . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)}))
18	17	rgen 3060	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
19		peano5 7255	. . . 4 ⊢ ((∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)})
20	6, 18, 19	mp2an 710	. . 3 ⊢ ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)}
21	20	sseli 3740	. 2 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
22		abid 2748	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜑))
23	21, 22	sylib 208	1 ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ∈ wcel 2139  {cab 2746  ∀wral 3050   ⊆ wss 3715  ∅c0 4058  suc csuc 5886  ωcom 7231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-om 7232

This theorem is referenced by:  finds1  7261  onnseq  7611  nnacl  7862  nnmcl  7863  nnecl  7864  nnacom  7868  nnaass  7873  nndi  7874  nnmass  7875  nnmsucr  7876  nnmcom  7877  nnmordi  7882  omsmolem  7904  isinf  8340  unblem2  8380  fiint  8404  dffi3  8504  card2inf  8627  cantnfle  8743  cantnflt  8744  cantnflem1  8761  cnfcom  8772  trcl  8779  fseqenlem1  9057  infpssrlem3  9339  fin23lem26  9359  axdc3lem2  9485  axdc4lem  9489  axdclem2  9554  wunr1om  9753  wuncval2  9781  tskr1om  9801  grothomex  9863  peano5nni  11235  neibastop2lem  32682  finxpreclem6  33562