Theorem tfinds3 7807

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 3161	. . 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 242	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑)))
16	2, 4, 6, 8, 9, 11, 15	tfinds 7802	1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∅c0 4285  Oncon0 6317  Lim wlim 6318  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323

This theorem is referenced by:  oacl  8462  omcl  8463  oecl  8464  oawordri  8477  oaass  8488  oarec  8489  omordi  8493  omwordri  8499  odi  8506  omass  8507  oen0  8514  oewordri  8520  oeworde  8521  oeoelem  8526  omabs  8579  tfindsd  44461