Theorem findes 4512

Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)

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

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2907	. 2 ⊢ (𝑧 = ∅ → ([𝑧 / 𝑥]𝜑 ↔ [∅ / 𝑥]𝜑))
2		sbequ 1812	. 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3		dfsbcq2 2907	. 2 ⊢ (𝑧 = suc 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑))
4		sbequ12r 1745	. 2 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑 ↔ 𝜑))
5		findes.1	. 2 ⊢ [∅ / 𝑥]𝜑
6		nfv 1508	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ ω
7		nfs1v 1910	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
8		nfsbc1v 2922	. . . . 5 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑
9	7, 8	nfim 1551	. . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)
10	6, 9	nfim 1551	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))
11		eleq1 2200	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
12		sbequ12 1744	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13		suceq 4319	. . . . . 6 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
14		dfsbcq 2906	. . . . . 6 ⊢ (suc 𝑥 = suc 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑))
15	13, 14	syl 14	. . . . 5 ⊢ (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑))
16	12, 15	imbi12d 233	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)))
17	11, 16	imbi12d 233	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ω → (𝜑 → [suc 𝑥 / 𝑥]𝜑)) ↔ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))))
18		findes.2	. . 3 ⊢ (𝑥 ∈ ω → (𝜑 → [suc 𝑥 / 𝑥]𝜑))
19	10, 17, 18	chvar 1730	. 2 ⊢ (𝑦 ∈ ω → ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))
20	1, 2, 3, 4, 5, 19	finds 4509	1 ⊢ (𝑥 ∈ ω → 𝜑)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  [wsb 1735  [wsbc 2904  ∅c0 3358  suc csuc 4282  ωcom 4499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-suc 4288  df-iom 4500

This theorem is referenced by: (None)