Theorem tfindes 2404

Description: Transfinite Induction with explicit substitution. The first hypothesis is the basis, the second is the induction hypothesis for successors, and the third is the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197.

Hypotheses

Ref	Expression
tfindes.1	⊢ [∅ / x]φ
tfindes.2	⊢ (x ∈ On → (φ → [suc x / x]φ))
tfindes.3	⊢ (Lim y → (∀x ∈ y φ → [y / x]φ))

Assertion

Ref	Expression
tfindes	⊢ (x ∈ On → φ)

Distinct variable group(s):   x,y   φ,y

Proof of Theorem tfindes

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. 2 ⊢ (y = ∅ → ([y / x]φ ↔ [∅ / x]φ))
2		sbequ 877	. 2 ⊢ (y = z → ([y / x]φ ↔ [z / x]φ))
3		dfsbcq 1442	. 2 ⊢ (y = suc z → ([y / x]φ ↔ [suc z / x]φ))
4		sbequ12r 866	. 2 ⊢ (y = x → ([y / x]φ ↔ φ))
5		tfindes.1	. 2 ⊢ [∅ / x]φ
6		ax-17 925	. . . 4 ⊢ (z ∈ On → ∀x z ∈ On)
7		hbs1 986	. . . . 5 ⊢ ([z / x]φ → ∀x[z / x]φ)
8		visset 1350	. . . . . . 7 ⊢ z ∈ V
9	8	sucex 2303	. . . . . 6 ⊢ suc z ∈ V
10	9	hbsbcv 1447	. . . . 5 ⊢ ([suc z / x]φ → ∀x[suc z / x]φ)
11	7, 10	hbim 702	. . . 4 ⊢ (([z / x]φ → [suc z / x]φ) → ∀x([z / x]φ → [suc z / x]φ))
12	6, 11	hbim 702	. . 3 ⊢ ((z ∈ On → ([z / x]φ → [suc z / x]φ)) → ∀x(z ∈ On → ([z / x]φ → [suc z / x]φ)))
13		eleq1 1149	. . . 4 ⊢ (x = z → (x ∈ On ↔ z ∈ On))
14		sbequ12 865	. . . . 5 ⊢ (x = z → (φ ↔ [z / x]φ))
15		suceq 2288	. . . . . 6 ⊢ (x = z → suc x = suc z)
16		dfsbcq 1442	. . . . . 6 ⊢ (suc x = suc z → ([suc x / x]φ ↔ [suc z / x]φ))
17	15, 16	syl 12	. . . . 5 ⊢ (x = z → ([suc x / x]φ ↔ [suc z / x]φ))
18	14, 17	imbi12d 474	. . . 4 ⊢ (x = z → ((φ → [suc x / x]φ) ↔ ([z / x]φ → [suc z / x]φ)))
19	13, 18	imbi12d 474	. . 3 ⊢ (x = z → ((x ∈ On → (φ → [suc x / x]φ)) ↔ (z ∈ On → ([z / x]φ → [suc z / x]φ))))
20		tfindes.2	. . 3 ⊢ (x ∈ On → (φ → [suc x / x]φ))
21	12, 19, 20	chv2 850	. 2 ⊢ (z ∈ On → ([z / x]φ → [suc z / x]φ))
22		tfindes.3	. . 3 ⊢ (Lim y → (∀x ∈ y φ → [y / x]φ))
23		ax-17 925	. . . 4 ⊢ (φ → ∀zφ)
24	23, 7, 14	cbvral 1331	. . 3 ⊢ (∀x ∈ y φ ↔ ∀z ∈ y [z / x]φ)
25	22, 24	syl5ibr 182	. 2 ⊢ (Lim y → (∀z ∈ y [z / x]φ → [y / x]φ))
26	1, 2, 3, 4, 5, 21, 25	tfinds 2401	1 ⊢ (x ∈ On → φ)

Syntax hints:   → wi 2   ↔ wb 127   = weq 797  [wsb 852   = wceq 1091   ∈ wcel 1092  ∀wral 1201  [wsbc 1440  ∅c0 1707  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  tfinds2 2405

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205

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