Theorem tfinds2 7843

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 5265	. . . 4 ⊢ ∅ ∈ V
3		tfinds2.1	. . . . 5 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 340	. . . 4 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	sbcie 3798	. . 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 3771	. . . . . 6 ⊢ (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))))
10	9	elv 3455	. . . . 5 ⊢ [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃)))
11		sbcimg 3805	. . . . . 6 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)))))
12	11	elv 3455	. . . . 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 3822	. . . 4 ⊢ ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
15		sbcimg 3805	. . . . 5 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)) ↔ ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃))))
16	15	elv 3455	. . . 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 3454	. . . 4 ⊢ 𝑥 ∈ V
19		tfinds2.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
20	19	bicomd 223	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))
21	20	equcoms 2020	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜒 ↔ 𝜑))
22	21	imbi2d 340	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝜏 → 𝜒) ↔ (𝜏 → 𝜑)))
23	18, 22	sbcie 3798	. . 3 ⊢ ([𝑥 / 𝑦](𝜏 → 𝜒) ↔ (𝜏 → 𝜑))
24		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
25	24	sucex 7785	. . . . . 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 3798	. . . . 5 ⊢ ([suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ (𝜏 → 𝜃))
29	28	sbcbii 3813	. . . 4 ⊢ ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ [𝑥 / 𝑦](𝜏 → 𝜃))
30		suceq 6403	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
31	30	sbcco2 3783	. . . 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 3328	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 (𝜏 → 𝜑) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒))
36		sbsbc 3760	. . . 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 3159	. . . . . . . 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 3771	. . . . . 6 ⊢ (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑))))
43	42	elv 3455	. . . . 5 ⊢ [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
44		sbcimg 3805	. . . . . 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 3455	. . . . 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 6347	. . . . 5 ⊢ (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
48	24, 47	sbcie 3798	. . . 4 ⊢ ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
49		sbcimg 3805	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)) ↔ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑))))
50	49	elv 3455	. . . 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 7842	1 ⊢ (𝑥 ∈ On → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  [wsb 2065   ∈ wcel 2109  ∀wral 3045  Vcvv 3450  [wsbc 3756  ∅c0 4299  Oncon0 6335  Lim wlim 6336  suc csuc 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341

This theorem is referenced by:  inar1  10735  grur1a  10779