Theorem tfinds3 7857

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 3190	. . 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 7852	1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∅c0 4286  Oncon0 6360  Lim wlim 6361  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366

This theorem is referenced by:  oacl  8516  omcl  8517  oecl  8518  oawordri  8531  oaass  8542  oarec  8543  omordi  8547  omwordri  8553  odi  8560  omass  8561  oen0  8568  oewordri  8574  oeworde  8575  oeoelem  8580  omabs  8633  tfindsd  44954