Theorem tfinds2 7885

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 5313	. . . 4 ⊢ ∅ ∈ V
3		tfinds2.1	. . . . 5 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
4	3	imbi2d 340	. . . 4 ⊢ (𝑥 = ∅ → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓)))
5	2, 4	sbcie 3835	. . 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 3806	. . . . . 6 ⊢ (𝑥 ∈ V → [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))))
10	9	elv 3483	. . . . 5 ⊢ [𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃)))
11		sbcimg 3843	. . . . . 6 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦](𝑦 ∈ On → ((𝜏 → 𝜒) → (𝜏 → 𝜃))) ↔ ([𝑥 / 𝑦]𝑦 ∈ On → [𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)))))
12	11	elv 3483	. . . . 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 3862	. . . 4 ⊢ ([𝑥 / 𝑦]𝑦 ∈ On ↔ 𝑥 ∈ On)
15		sbcimg 3843	. . . . 5 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑦]((𝜏 → 𝜒) → (𝜏 → 𝜃)) ↔ ([𝑥 / 𝑦](𝜏 → 𝜒) → [𝑥 / 𝑦](𝜏 → 𝜃))))
16	15	elv 3483	. . . 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 3482	. . . 4 ⊢ 𝑥 ∈ V
19		tfinds2.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
20	19	bicomd 223	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑))
21	20	equcoms 2017	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜒 ↔ 𝜑))
22	21	imbi2d 340	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝜏 → 𝜒) ↔ (𝜏 → 𝜑)))
23	18, 22	sbcie 3835	. . 3 ⊢ ([𝑥 / 𝑦](𝜏 → 𝜒) ↔ (𝜏 → 𝜑))
24		vex 3482	. . . . . . 7 ⊢ 𝑦 ∈ V
25	24	sucex 7826	. . . . . 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 3835	. . . . 5 ⊢ ([suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ (𝜏 → 𝜃))
29	28	sbcbii 3852	. . . 4 ⊢ ([𝑥 / 𝑦][suc 𝑦 / 𝑥](𝜏 → 𝜑) ↔ [𝑥 / 𝑦](𝜏 → 𝜃))
30		suceq 6452	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
31	30	sbcco2 3818	. . . 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 3356	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 (𝜏 → 𝜑) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒))
36		sbsbc 3795	. . . 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 3178	. . . . . . . 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 3806	. . . . . 6 ⊢ (𝑦 ∈ V → [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑))))
43	42	elv 3483	. . . . 5 ⊢ [𝑦 / 𝑥](Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)))
44		sbcimg 3843	. . . . . 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 3483	. . . . 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 6398	. . . . 5 ⊢ (𝑥 = 𝑦 → (Lim 𝑥 ↔ Lim 𝑦))
48	24, 47	sbcie 3835	. . . 4 ⊢ ([𝑦 / 𝑥]Lim 𝑥 ↔ Lim 𝑦)
49		sbcimg 3843	. . . . 5 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥](∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → (𝜏 → 𝜑)) ↔ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 (𝜏 → 𝜒) → [𝑦 / 𝑥](𝜏 → 𝜑))))
50	49	elv 3483	. . . 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 7884	1 ⊢ (𝑥 ∈ On → (𝜏 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  [wsb 2062   ∈ wcel 2106  ∀wral 3059  Vcvv 3478  [wsbc 3791  ∅c0 4339  Oncon0 6386  Lim wlim 6387  suc csuc 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392

This theorem is referenced by:  inar1  10813  grur1a  10857