Theorem finds1 7900

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 2762	. 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 7899	. 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
10	1, 9	mpi 21	1 ⊢ (𝑥 ∈ ω → 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∅c0 4282  suc csuc 6363  ωcom 7866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-om 7867

This theorem is used by:  findcard  9162  findcard2  9163  alephfplem3  10113  pwsdompw  10209  hsmexlem4  10435