Theorem finds1 7897

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 2763	. 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 11	. . 3 ⊢ (∅ = ∅ → 𝜓)
7		finds1.5	. . . 4 ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))
8	7	a1d 26	. . 3 ⊢ (𝑦 ∈ ω → (∅ = ∅ → (𝜒 → 𝜃)))
9	2, 3, 4, 6, 8	finds2 7896	. 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
10	1, 9	mpi 21	1 ⊢ (𝑥 ∈ ω → 𝜑)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∅c0 4287  suc csuc 6364  ωcom 7863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-om 7864

This theorem is referenced by:  findcard  9149  findcard2  9150  alephfplem3  10091  pwsdompw  10187  hsmexlem4  10414