Theorem finds1 7875

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 2729	. 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 25	. . 3 ⊢ (𝑦 ∈ ω → (∅ = ∅ → (𝜒 → 𝜃)))
9	2, 3, 4, 6, 8	finds2 7874	. 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
10	1, 9	mpi 20	1 ⊢ (𝑥 ∈ ω → 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∅c0 4296  suc csuc 6334  ωcom 7842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-om 7843

This theorem is referenced by:  findcard  9127  findcard2  9128  alephfplem3  10059  pwsdompw  10156  hsmexlem4  10382