Theorem tfinds2 7901

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 5325	. . . 4 ⊢ ∅ ∈ V
3		tfinds2.1	. . . . 5 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 340	. . . 4 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	sbcie 3848	. . 3 ⊢ ([∅ / 𝑥](𝜏 → 𝜑) ↔ (𝜏 → 𝜓))
6	1, 5	mpbir 231	. 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 3819	. . . . . 6 ⊢ (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))))
10	9	elv 3493	. . . . 5 ⊢ [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃)))
11		sbcimg 3856	. . . . . 6 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)))))
12	11	elv 3493	. . . . 5 ⊢ ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃))))
13	10, 12	mpbi 230	. . . 4 ⊢ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)))
14		sbcel1v 3875	. . . 4 ⊢ ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
15		sbcimg 3856	. . . . 5 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)) ↔ ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃))))
16	15	elv 3493	. . . 4 ⊢ ([𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)) ↔ ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃)))
17	13, 14, 16	3imtr3i 291	. . 3 ⊢ (𝑥 ∈ On → ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃)))
18		vex 3492	. . . 4 ⊢ 𝑥 ∈ V
19		tfinds2.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
20	19	bicomd 223	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))
21	20	equcoms 2019	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜒 ↔ 𝜑))
22	21	imbi2d 340	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝜏 → 𝜒) ↔ (𝜏 → 𝜑)))
23	18, 22	sbcie 3848	. . 3 ⊢ ([𝑥 / 𝑦](𝜏 → 𝜒) ↔ (𝜏 → 𝜑))
24		vex 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
25	24	sucex 7842	. . . . . 6 ⊢ suc 𝑦 ∈ V
26		tfinds2.3	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
27	26	imbi2d 340	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃)))
28	25, 27	sbcie 3848	. . . . 5 ⊢ ([suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ (𝜏 → 𝜃))
29	28	sbcbii 3865	. . . 4 ⊢ ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ [𝑥 / 𝑦](𝜏 → 𝜃))
30		suceq 6461	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
31	30	sbcco2 3831	. . . 4 ⊢ ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ [suc 𝑥 / 𝑥](𝜏 → 𝜑))
32	29, 31	bitr3i 277	. . 3 ⊢ ([𝑥 / 𝑦](𝜏 → 𝜃) ↔ [suc 𝑥 / 𝑥](𝜏 → 𝜑))
33	17, 23, 32	3imtr3g 295	. 2 ⊢ (𝑥 ∈ On → ((𝜏 → 𝜑) → [suc 𝑥 / 𝑥](𝜏 → 𝜑)))
34	19	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒)))
35	34	sbralie 3366	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 (𝜏 → 𝜑) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒))
36		sbsbc 3808	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒))
37	35, 36	bitr2i 276	. . 3 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) ↔ ∀𝑥 ∈ 𝑦 (𝜏 → 𝜑))
38		r19.21v 3186	. . . . . . . 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	biimtrid 242	. . . . . . 7 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
42	41	sbcth 3819	. . . . . 6 ⊢ (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑))))
43	42	elv 3493	. . . . 5 ⊢ [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
44		sbcimg 3856	. . . . . 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 3493	. . . . 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 230	. . . 4 ⊢ ([𝑦 / 𝑥]Lim 𝑥 → [𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
47		limeq 6407	. . . . 5 ⊢ (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
48	24, 47	sbcie 3848	. . . 4 ⊢ ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
49		sbcimg 3856	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)) ↔ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑))))
50	49	elv 3493	. . . 4 ⊢ ([𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)) ↔ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑)))
51	46, 48, 50	3imtr3i 291	. . 3 ⊢ (Lim 𝑦 → ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑)))
52	37, 51	biimtrrid 243	. 2 ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 (𝜏 → 𝜑) → [𝑦 / 𝑥](𝜏 → 𝜑)))
53	6, 33, 52	tfindes 7900	1 ⊢ (𝑥 ∈ On → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  [wsb 2064   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  [wsbc 3804  ∅c0 4352  Oncon0 6395  Lim wlim 6396  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401

This theorem is referenced by:  inar1  10844  grur1a  10888