Theorem bj-findes 15451

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15449 for explanations. From this version, it is easy to prove findes 4633. (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 1539	. . . 4 ⊢ Ⅎ𝑦𝜑
2		nfv 1539	. . . 4 ⊢ Ⅎ𝑦[suc 𝑥 / 𝑥]𝜑
3	1, 2	nfim 1583	. . 3 ⊢ Ⅎ𝑦(𝜑 → [suc 𝑥 / 𝑥]𝜑)
4		nfs1v 1955	. . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
5		nfsbc1v 3004	. . . 4 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑
6	4, 5	nfim 1583	. . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)
7		sbequ12 1782	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8		suceq 4431	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
9	8	sbceq1d 2990	. . . 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 2722	. 2 ⊢ (∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))
12		nfsbc1v 3004	. . 3 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
13		sbceq1a 2995	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
14	13	biimprd 158	. . 3 ⊢ (𝑥 = ∅ → ([∅ / 𝑥]𝜑 → 𝜑))
15		sbequ1 1779	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
16		sbceq1a 2995	. . . 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 15449	. 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 1364  [wsb 1773  ∀wral 2472  [wsbc 2985  ∅c0 3446  suc csuc 4394  ωcom 4620

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4155  ax-pr 4238  ax-un 4462  ax-setind 4567  ax-bd0 15283  ax-bdim 15284  ax-bdan 15285  ax-bdor 15286  ax-bdn 15287  ax-bdal 15288  ax-bdex 15289  ax-bdeq 15290  ax-bdel 15291  ax-bdsb 15292  ax-bdsep 15354  ax-infvn 15411

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4400  df-iom 4621  df-bdc 15311  df-bj-ind 15397

This theorem is referenced by: (None)