Theorem bj-bdfindes 15595

Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 15593 for explanations. From this version, it is easy to prove the bounded version of findes 4639. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-bdfindes.bd	⊢ BOUNDED 𝜑

Assertion

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

Proof of Theorem bj-bdfindes

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1542	. . . 4 ⊢ Ⅎ𝑦𝜑
2		nfv 1542	. . . 4 ⊢ Ⅎ𝑦[suc 𝑥 / 𝑥]𝜑
3	1, 2	nfim 1586	. . 3 ⊢ Ⅎ𝑦(𝜑 → [suc 𝑥 / 𝑥]𝜑)
4		nfs1v 1958	. . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
5		nfsbc1v 3008	. . . 4 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑
6	4, 5	nfim 1586	. . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)
7		sbequ12 1785	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8		suceq 4437	. . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
9	8	sbceq1d 2994	. . . 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 2725	. 2 ⊢ (∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))
12		bj-bdfindes.bd	. . 3 ⊢ BOUNDED 𝜑
13		nfsbc1v 3008	. . 3 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑
14		sbceq1a 2999	. . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑))
15	14	biimprd 158	. . 3 ⊢ (𝑥 = ∅ → ([∅ / 𝑥]𝜑 → 𝜑))
16		sbequ1 1782	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
17		sbceq1a 2999	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ [suc 𝑦 / 𝑥]𝜑))
18	17	biimprd 158	. . 3 ⊢ (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑 → 𝜑))
19	12, 13, 4, 5, 15, 16, 18	bj-bdfindis 15593	. 2 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
20	11, 19	sylan2b 287	1 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  [wsb 1776  ∀wral 2475  [wsbc 2989  ∅c0 3450  suc csuc 4400  ωcom 4626  BOUNDED wbd 15458

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4159  ax-pr 4242  ax-un 4468  ax-bd0 15459  ax-bdor 15462  ax-bdex 15465  ax-bdeq 15466  ax-bdel 15467  ax-bdsb 15468  ax-bdsep 15530  ax-infvn 15587

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-iom 4627  df-bdc 15487  df-bj-ind 15573

This theorem is referenced by: (None)