Theorem finds1 4511

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, 22-Mar-2006.)

Hypotheses

Ref	Expression
finds1.1	⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
finds1.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
finds1.3	⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
finds1.4	⊢ 𝜓
finds1.5	⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))

Assertion

Ref	Expression
finds1	⊢ (𝑥 ∈ ω → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)

Proof of Theorem finds1

Step	Hyp	Ref	Expression
1		eqid 2137	. 2 ⊢ ∅ = ∅
2		finds1.1	. . 3 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
3		finds1.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
4		finds1.3	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
5		finds1.4	. . . 4 ⊢ 𝜓
6	5	a1i 9	. . 3 ⊢ (∅ = ∅ → 𝜓)
7		finds1.5	. . . 4 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))
8	7	a1d 22	. . 3 ⊢ (𝑦 ∈ ω → (∅ = ∅ → (𝜒 → 𝜃)))
9	2, 3, 4, 6, 8	finds2 4510	. 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
10	1, 9	mpi 15	1 ⊢ (𝑥 ∈ ω → 𝜑)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∅c0 3358  suc csuc 4282  ωcom 4499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-suc 4288  df-iom 4500

This theorem is referenced by:  findcard  6775  findcard2  6776  findcard2s  6777