Theorem finds2 4601

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 4131	. . . . . 6 ⊢ ∅ ∈ V
3		finds2.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 230	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	elab 2882	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜓))
6	1, 5	mpbir 146	. . . 4 ⊢ ∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)}
7		finds2.5	. . . . . . 7 ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃)))
8	7	a2d 26	. . . . . 6 ⊢ (𝑦 ∈ ω → ((𝜏 → 𝜒) → (𝜏 → 𝜃)))
9		vex 2741	. . . . . . 7 ⊢ 𝑦 ∈ V
10		finds2.2	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
11	10	imbi2d 230	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒)))
12	9, 11	elab 2882	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜒))
13	9	sucex 4499	. . . . . . 7 ⊢ suc 𝑦 ∈ V
14		finds2.3	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
15	14	imbi2d 230	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃)))
16	13, 15	elab 2882	. . . . . 6 ⊢ (suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜃))
17	8, 12, 16	3imtr4g 205	. . . . 5 ⊢ (𝑦 ∈ ω → (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)}))
18	17	rgen 2530	. . . 4 ⊢ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
19		peano5 4598	. . . 4 ⊢ ((∅ ∈ {𝑥 ∣ (𝜏 → 𝜑)} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)} → suc 𝑦 ∈ {𝑥 ∣ (𝜏 → 𝜑)})) → ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)})
20	6, 18, 19	mp2an 426	. . 3 ⊢ ω ⊆ {𝑥 ∣ (𝜏 → 𝜑)}
21	20	sseli 3152	. 2 ⊢ (𝑥 ∈ ω → 𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)})
22		abid 2165	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ (𝜏 → 𝜑)} ↔ (𝜏 → 𝜑))
23	21, 22	sylib 122	1 ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455   ⊆ wss 3130  ∅c0 3423  suc csuc 4366  ωcom 4590

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 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588

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 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-suc 4372  df-iom 4591

This theorem is referenced by:  finds1  4602  frecrdg  6409  nnacl  6481  nnmcl  6482  nnacom  6485  nnaass  6486  nndi  6487  nnmass  6488  nnmsucr  6489  nnmcom  6490  nnsucsssuc  6493  nntri3or  6494  nnaordi  6509  nnaword  6512  nnmordi  6517  nnaordex  6529  fiintim  6928  prarloclem3  7496  frec2uzuzd  10402  frec2uzrdg  10409