Theorem bj-findes 16751

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 16749 for explanations. From this version, it is easy to prove findes 4725. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-findes	⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)

Proof of Theorem bj-findes

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1577	. . . 4 ⊢ Ⅎ𝑦𝜑
2		nfv 1577	. . . 4 ⊢ Ⅎ𝑦[suc 𝑥 / 𝑥]𝜑
3	1, 2	nfim 1621	. . 3 ⊢ Ⅎ𝑦(𝜑 → [suc 𝑥 / 𝑥]𝜑)
4		nfs1v 1993	. . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
5		nfsbc1v 3061	. . . 4 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑
6	4, 5	nfim 1621	. . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)
7		sbequ12 1820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8		suceq 4523	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
9	8	sbceq1d 3047	. . . 4 ⊢ (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑))
10	7, 9	imbi12d 234	. . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)))
11	3, 6, 10	cbvral 2774	. 2 ⊢ (∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))
12		nfsbc1v 3061	. . 3 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
13		sbceq1a 3052	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
14	13	biimprd 158	. . 3 ⊢ (𝑥 = ∅ → ([∅ / 𝑥]𝜑 → 𝜑))
15		sbequ1 1817	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
16		sbceq1a 3052	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ [suc 𝑦 / 𝑥]𝜑))
17	16	biimprd 158	. . 3 ⊢ (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑 → 𝜑))
18	12, 4, 5, 14, 15, 17	bj-findis 16749	. 2 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
19	11, 18	sylan2b 287	1 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  [wsb 1811  ∀wral 2520  [wsbc 3042  ∅c0 3508  suc csuc 4486  ωcom 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-nul 4236  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-bd0 16583  ax-bdim 16584  ax-bdan 16585  ax-bdor 16586  ax-bdn 16587  ax-bdal 16588  ax-bdex 16589  ax-bdeq 16590  ax-bdel 16591  ax-bdsb 16592  ax-bdsep 16654  ax-infvn 16711

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-suc 4492  df-iom 4713  df-bdc 16611  df-bj-ind 16697

This theorem is referenced by: (None)