Theorem findes 2400

Description: Finite induction with explicit substitution. The first hypothesis is the basis, and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by Raph Levien, 9-Jul-03.)

Hypotheses

Ref	Expression
findes.1	⊢ [∅ / x]φ
findes.2	⊢ (x ∈ ω → (φ → [suc x / x]φ))

Assertion

Ref	Expression
findes	⊢ (x ∈ ω → φ)

Proof of Theorem findes

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. 2 ⊢ (z = ∅ → ([z / x]φ ↔ [∅ / x]φ))
2		sbequ 877	. 2 ⊢ (z = y → ([z / x]φ ↔ [y / x]φ))
3		dfsbcq 1442	. 2 ⊢ (z = suc y → ([z / x]φ ↔ [suc y / x]φ))
4		sbequ12r 866	. 2 ⊢ (z = x → ([z / x]φ ↔ φ))
5		findes.1	. 2 ⊢ [∅ / x]φ
6		ax-17 925	. . . 4 ⊢ (y ∈ ω → ∀x y ∈ ω)
7		hbs1 986	. . . . 5 ⊢ ([y / x]φ → ∀x[y / x]φ)
8		visset 1350	. . . . . . 7 ⊢ y ∈ V
9	8	sucex 2303	. . . . . 6 ⊢ suc y ∈ V
10	9	hbsbcv 1447	. . . . 5 ⊢ ([suc y / x]φ → ∀x[suc y / x]φ)
11	7, 10	hbim 702	. . . 4 ⊢ (([y / x]φ → [suc y / x]φ) → ∀x([y / x]φ → [suc y / x]φ))
12	6, 11	hbim 702	. . 3 ⊢ ((y ∈ ω → ([y / x]φ → [suc y / x]φ)) → ∀x(y ∈ ω → ([y / x]φ → [suc y / x]φ)))
13		eleq1 1149	. . . 4 ⊢ (x = y → (x ∈ ω ↔ y ∈ ω))
14		sbequ12 865	. . . . 5 ⊢ (x = y → (φ ↔ [y / x]φ))
15		suceq 2288	. . . . . 6 ⊢ (x = y → suc x = suc y)
16		dfsbcq 1442	. . . . . 6 ⊢ (suc x = suc y → ([suc x / x]φ ↔ [suc y / x]φ))
17	15, 16	syl 12	. . . . 5 ⊢ (x = y → ([suc x / x]φ ↔ [suc y / x]φ))
18	14, 17	imbi12d 474	. . . 4 ⊢ (x = y → ((φ → [suc x / x]φ) ↔ ([y / x]φ → [suc y / x]φ)))
19	13, 18	imbi12d 474	. . 3 ⊢ (x = y → ((x ∈ ω → (φ → [suc x / x]φ)) ↔ (y ∈ ω → ([y / x]φ → [suc y / x]φ))))
20		findes.2	. . 3 ⊢ (x ∈ ω → (φ → [suc x / x]φ))
21	12, 19, 20	chv2 850	. 2 ⊢ (y ∈ ω → ([y / x]φ → [suc y / x]φ))
22	1, 2, 3, 4, 5, 21	finds 2397	1 ⊢ (x ∈ ω → φ)

Syntax hints:   → wi 2   ↔ wb 127   = weq 797  [wsb 852   = wceq 1091   ∈ wcel 1092  [wsbc 1440  ∅c0 1707  suc csuc 2201  ωcom 2372

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-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  df-om 2373

metamath.org