Theorem tfinds3 7862

Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)

Hypotheses

Ref	Expression
tfinds3.1	⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
tfinds3.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
tfinds3.3	⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
tfinds3.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
tfinds3.5	⊢ (𝜂 → 𝜓)
tfinds3.6	⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃)))
tfinds3.7	⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))

Assertion

Ref	Expression
tfinds3	⊢ (𝐴 ∈ On → (𝜂 → 𝜏))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑦   𝜒,𝑥   𝜏,𝑥   𝑥,𝑦,𝜂

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

Proof of Theorem tfinds3

Step	Hyp	Ref	Expression
1		tfinds3.1	. . 3 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
2	1	imbi2d 343	. 2 ⊢ (𝑥 = ∅ → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜓)))
3		tfinds3.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
4	3	imbi2d 343	. 2 ⊢ (𝑥 = 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜒)))
5		tfinds3.3	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
6	5	imbi2d 343	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜃)))
7		tfinds3.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
8	7	imbi2d 343	. 2 ⊢ (𝑥 = 𝐴 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜏)))
9		tfinds3.5	. 2 ⊢ (𝜂 → 𝜓)
10		tfinds3.6	. . 3 ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃)))
11	10	a2d 30	. 2 ⊢ (𝑦 ∈ On → ((𝜂 → 𝜒) → (𝜂 → 𝜃)))
12		r19.21v 3187	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) ↔ (𝜂 → ∀𝑦 ∈ 𝑥 𝜒))
13		tfinds3.7	. . . 4 ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
14	13	a2d 30	. . 3 ⊢ (Lim 𝑥 → ((𝜂 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜂 → 𝜑)))
15	12, 14	biimtrid 245	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑)))
16	2, 4, 6, 8, 9, 11, 15	tfinds 7857	1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∅c0 4279  Oncon0 6357  Lim wlim 6358  suc csuc 6359

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

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-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363

This theorem is used by:  oacl  8525  omcl  8526  oecl  8527  oawordri  8540  oaass  8551  oarec  8552  omordi  8556  omwordri  8562  odi  8569  omass  8570  oen0  8577  oewordri  8583  oeworde  8584  oeoelem  8589  omabs  8642  tfindsd  45051