Theorem tfinds3 7806

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 340	. 2 ⊢ (𝑥 = ∅ → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜓)))
3		tfinds3.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
4	3	imbi2d 340	. 2 ⊢ (𝑥 = 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜒)))
5		tfinds3.3	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
6	5	imbi2d 340	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜃)))
7		tfinds3.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
8	7	imbi2d 340	. 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 29	. 2 ⊢ (𝑦 ∈ On → ((𝜂 → 𝜒) → (𝜂 → 𝜃)))
12		r19.21v 3172	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) ↔ (𝜂 → ∀𝑦 ∈ 𝑥 𝜒))
13		tfinds3.7	. . . 4 ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
14	13	a2d 29	. . 3 ⊢ (Lim 𝑥 → ((𝜂 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜂 → 𝜑)))
15	12, 14	biimtrid 241	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑)))
16	2, 4, 6, 8, 9, 11, 15	tfinds 7801	1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∅c0 4287  Oncon0 6322  Lim wlim 6323  suc csuc 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328

This theorem is referenced by:  oacl  8486  omcl  8487  oecl  8488  oawordri  8502  oaass  8513  oarec  8514  omordi  8518  omwordri  8524  odi  8531  omass  8532  oen0  8538  oewordri  8544  oeworde  8545  oeoelem  8550  omabs  8602  tfindsd  42607