Theorem tfinds2 7845

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 5258	. . . 4 ⊢ ∅ ∈ V
3		tfinds2.1	. . . . 5 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 342	. . . 4 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	sbcie 3786	. . 3 ⊢ ([∅ / 𝑥](𝜏 → 𝜑) ↔ (𝜏 → 𝜓))
6	1, 5	mpbir 233	. 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 3760	. . . . . 6 ⊢ (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))))
10	9	elv 3460	. . . . 5 ⊢ [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃)))
11		sbcimg 3793	. . . . . 6 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)))))
12	11	elv 3460	. . . . 5 ⊢ ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃))))
13	10, 12	mpbi 232	. . . 4 ⊢ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)))
14		sbcel1v 3810	. . . 4 ⊢ ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
15		sbcimg 3793	. . . . 5 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)) ↔ ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃))))
16	15	elv 3460	. . . 4 ⊢ ([𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)) ↔ ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃)))
17	13, 14, 16	3imtr3i 293	. . 3 ⊢ (𝑥 ∈ On → ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃)))
18		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
19		tfinds2.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
20	19	bicomd 225	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))
21	20	equcoms 2041	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜒 ↔ 𝜑))
22	21	imbi2d 342	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝜏 → 𝜒) ↔ (𝜏 → 𝜑)))
23	18, 22	sbcie 3786	. . 3 ⊢ ([𝑥 / 𝑦](𝜏 → 𝜒) ↔ (𝜏 → 𝜑))
24		vex 3459	. . . . . . 7 ⊢ 𝑦 ∈ V
25	24	sucex 7790	. . . . . 6 ⊢ suc 𝑦 ∈ V
26		tfinds2.3	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
27	26	imbi2d 342	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜃)))
28	25, 27	sbcie 3786	. . . . 5 ⊢ ([suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ (𝜏 → 𝜃))
29	28	sbcbii 3801	. . . 4 ⊢ ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ [𝑥 / 𝑦](𝜏 → 𝜃))
30		suceq 6415	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
31	30	sbcco2 3772	. . . 4 ⊢ ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ [suc 𝑥 / 𝑥](𝜏 → 𝜑))
32	29, 31	bitr3i 279	. . 3 ⊢ ([𝑥 / 𝑦](𝜏 → 𝜃) ↔ [suc 𝑥 / 𝑥](𝜏 → 𝜑))
33	17, 23, 32	3imtr3g 297	. 2 ⊢ (𝑥 ∈ On → ((𝜏 → 𝜑) → [suc 𝑥 / 𝑥](𝜏 → 𝜑)))
34	19	imbi2d 342	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜒)))
35	34	sbralie 3341	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 (𝜏 → 𝜑) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒))
36		sbsbc 3749	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒))
37	35, 36	bitr2i 278	. . 3 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) ↔ ∀𝑥 ∈ 𝑦 (𝜏 → 𝜑))
38		r19.21v 3188	. . . . . . . 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 244	. . . . . . 7 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
42	41	sbcth 3760	. . . . . 6 ⊢ (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑))))
43	42	elv 3460	. . . . 5 ⊢ [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
44		sbcimg 3793	. . . . . 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 3460	. . . . 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 232	. . . 4 ⊢ ([𝑦 / 𝑥]Lim 𝑥 → [𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
47		limeq 6359	. . . . 5 ⊢ (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
48	24, 47	sbcie 3786	. . . 4 ⊢ ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
49		sbcimg 3793	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)) ↔ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑))))
50	49	elv 3460	. . . 4 ⊢ ([𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)) ↔ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑)))
51	46, 48, 50	3imtr3i 293	. . 3 ⊢ (Lim 𝑦 → ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑)))
52	37, 51	biimtrrid 245	. 2 ⊢ (Lim 𝑦 → (∀𝑥 ∈ 𝑦 (𝜏 → 𝜑) → [𝑦 / 𝑥](𝜏 → 𝜑)))
53	6, 33, 52	tfindes 7844	1 ⊢ (𝑥 ∈ On → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1561  [wsb 2091   ∈ wcel 2143  ∀wral 3077  Vcvv 3455  [wsbc 3745  ∅c0 4286  Oncon0 6347  Lim wlim 6348  suc csuc 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-tr 5209  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353

This theorem is referenced by:  inar1  10734  grur1a  10778