Theorem finds1 7614

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 2824	. 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 7613	. 2 ⊢ (𝑥 ∈ ω → (∅ = ∅ → 𝜑))
10	1, 9	mpi 20	1 ⊢ (𝑥 ∈ ω → 𝜑)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ∈ wcel 2113  ∅c0 4294  suc csuc 6196  ωcom 7583

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-tr 5176  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-om 7584

This theorem is referenced by:  findcard  8760  findcard2  8761  pwfi  8822  alephfplem3  9535  pwsdompw  9629  hsmexlem4  9854