Theorem tfinds2 7558

Description: Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff 𝜏 is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)

Hypotheses

Ref	Expression
tfinds2.1	⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
tfinds2.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
tfinds2.3	⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
tfinds2.4	⊢ (𝜏 → 𝜓)
tfinds2.5	⊢ (𝑦 ∈ On → (𝜏 → (𝜒 → 𝜃)))
tfinds2.6	⊢ (Lim 𝑥 → (𝜏 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))

Assertion

Ref	Expression
tfinds2	⊢ (𝑥 ∈ On → (𝜏 → 𝜑))

Distinct variable groups:   𝑥,𝑦,𝜏   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜑,𝑦

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

Proof of Theorem tfinds2

Step	Hyp	Ref	Expression
1		tfinds2.4	. . 3 ⊢ (𝜏 → 𝜓)
2		0ex 5175	. . . 4 ⊢ ∅ ∈ V
3		tfinds2.1	. . . . 5 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 344	. . . 4 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	sbcie 3760	. . 3 ⊢ ([∅ / 𝑥](𝜏 → 𝜑) ↔ (𝜏 → 𝜓))
6	1, 5	mpbir 234	. 2 ⊢ [∅ / 𝑥](𝜏 → 𝜑)
7		tfinds2.5	. . . . . . . 8 ⊢ (𝑦 ∈ On → (𝜏 → (𝜒 → 𝜃)))
8	7	a2d 29	. . . . . . 7 ⊢ (𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃)))
9	8	sbcth 3735	. . . . . 6 ⊢ (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))))
10	9	elv 3446	. . . . 5 ⊢ [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃)))
11		sbcimg 3767	. . . . . 6 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)))))
12	11	elv 3446	. . . . 5 ⊢ ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃))))
13	10, 12	mpbi 233	. . . 4 ⊢ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)))
14		sbcel1v 3786	. . . 4 ⊢ ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
15		sbcimg 3767	. . . . 5 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)) ↔ ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃))))
16	15	elv 3446	. . . 4 ⊢ ([𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)) ↔ ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃)))
17	13, 14, 16	3imtr3i 294	. . 3 ⊢ (𝑥 ∈ On → ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃)))
18		vex 3444	. . . 4 ⊢ 𝑥 ∈ V
19		tfinds2.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
20	19	bicomd 226	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))
21	20	equcoms 2027	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜒 ↔ 𝜑))
22	21	imbi2d 344	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝜏 → 𝜒) ↔ (𝜏 → 𝜑)))
23	18, 22	sbcie 3760	. . 3 ⊢ ([𝑥 / 𝑦](𝜏 → 𝜒) ↔ (𝜏 → 𝜑))
24		vex 3444	. . . . . . 7 ⊢ 𝑦 ∈ V
25	24	sucex 7506	. . . . . 6 ⊢ suc 𝑦 ∈ V
26		tfinds2.3	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
27	26	imbi2d 344	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃)))
28	25, 27	sbcie 3760	. . . . 5 ⊢ ([suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ (𝜏 → 𝜃))
29	28	sbcbii 3776	. . . 4 ⊢ ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ [𝑥 / 𝑦](𝜏 → 𝜃))
30		suceq 6224	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
31	30	sbcco2 3747	. . . 4 ⊢ ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ [suc 𝑥 / 𝑥](𝜏 → 𝜑))
32	29, 31	bitr3i 280	. . 3 ⊢ ([𝑥 / 𝑦](𝜏 → 𝜃) ↔ [suc 𝑥 / 𝑥](𝜏 → 𝜑))
33	17, 23, 32	3imtr3g 298	. 2 ⊢ (𝑥 ∈ On → ((𝜏 → 𝜑) → [suc 𝑥 / 𝑥](𝜏 → 𝜑)))
34	19	imbi2d 344	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒)))
35	34	sbralie 3418	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 (𝜏 → 𝜑) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒))
36		sbsbc 3724	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒))
37	35, 36	bitr2i 279	. . 3 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) ↔ ∀𝑥 ∈ 𝑦 (𝜏 → 𝜑))
38		r19.21v 3142	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) ↔ (𝜏 → ∀𝑦 ∈ 𝑥 𝜒))
39		tfinds2.6	. . . . . . . . 9 ⊢ (Lim 𝑥 → (𝜏 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
40	39	a2d 29	. . . . . . . 8 ⊢ (Lim 𝑥 → ((𝜏 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜏 → 𝜑)))
41	38, 40	syl5bi 245	. . . . . . 7 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
42	41	sbcth 3735	. . . . . 6 ⊢ (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑))))
43	42	elv 3446	. . . . 5 ⊢ [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
44		sbcimg 3767	. . . . . 6 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑))) ↔ ([𝑦 / 𝑥]Lim 𝑥 → [𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))))
45	44	elv 3446	. . . . 5 ⊢ ([𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑))) ↔ ([𝑦 / 𝑥]Lim 𝑥 → [𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑))))
46	43, 45	mpbi 233	. . . 4 ⊢ ([𝑦 / 𝑥]Lim 𝑥 → [𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
47		limeq 6171	. . . . 5 ⊢ (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
48	24, 47	sbcie 3760	. . . 4 ⊢ ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
49		sbcimg 3767	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)) ↔ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑))))
50	49	elv 3446	. . . 4 ⊢ ([𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)) ↔ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑)))
51	46, 48, 50	3imtr3i 294	. . 3 ⊢ (Lim 𝑦 → ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑)))
52	37, 51	syl5bir 246	. 2 ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 (𝜏 → 𝜑) → [𝑦 / 𝑥](𝜏 → 𝜑)))
53	6, 33, 52	tfindes 7557	1 ⊢ (𝑥 ∈ On → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  [wsb 2069   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  [wsbc 3720  ∅c0 4243  Oncon0 6159  Lim wlim 6160  suc csuc 6161

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165

This theorem is referenced by:  inar1  10186  grur1a  10230