Theorem bj-findes 15850

Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15848 for explanations. From this version, it is easy to prove findes 4650. (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 1550	. . . 4 ⊢ Ⅎ𝑦𝜑
2		nfv 1550	. . . 4 ⊢ Ⅎ𝑦[suc 𝑥 / 𝑥]𝜑
3	1, 2	nfim 1594	. . 3 ⊢ Ⅎ𝑦(𝜑 → [suc 𝑥 / 𝑥]𝜑)
4		nfs1v 1966	. . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
5		nfsbc1v 3016	. . . 4 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑
6	4, 5	nfim 1594	. . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)
7		sbequ12 1793	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8		suceq 4448	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
9	8	sbceq1d 3002	. . . 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 2733	. 2 ⊢ (∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))
12		nfsbc1v 3016	. . 3 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
13		sbceq1a 3007	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
14	13	biimprd 158	. . 3 ⊢ (𝑥 = ∅ → ([∅ / 𝑥]𝜑 → 𝜑))
15		sbequ1 1790	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
16		sbceq1a 3007	. . . 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 15848	. 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 1372  [wsb 1784  ∀wral 2483  [wsbc 2997  ∅c0 3459  suc csuc 4411  ωcom 4637

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-nul 4169  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-bd0 15682  ax-bdim 15683  ax-bdan 15684  ax-bdor 15685  ax-bdn 15686  ax-bdal 15687  ax-bdex 15688  ax-bdeq 15689  ax-bdel 15690  ax-bdsb 15691  ax-bdsep 15753  ax-infvn 15810

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885  df-suc 4417  df-iom 4638  df-bdc 15710  df-bj-ind 15796

This theorem is referenced by: (None)