Theorem bdsetindis 15017

Description: Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdsetindis.bd	⊢ BOUNDED 𝜑
bdsetindis.nf0	⊢ Ⅎ𝑥𝜓
bdsetindis.nf1	⊢ Ⅎ𝑥𝜒
bdsetindis.nf2	⊢ Ⅎ𝑦𝜑
bdsetindis.nf3	⊢ Ⅎ𝑦𝜓
bdsetindis.1	⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓))
bdsetindis.2	⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑))

Assertion

Ref	Expression
bdsetindis	⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑)

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem bdsetindis

Step	Hyp	Ref	Expression
1		nfcv 2329	. . . . 5 ⊢ Ⅎ𝑥𝑦
2		bdsetindis.nf0	. . . . 5 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfralxy 2525	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 𝜓
4		bdsetindis.nf1	. . . 4 ⊢ Ⅎ𝑥𝜒
5	3, 4	nfim 1582	. . 3 ⊢ Ⅎ𝑥(∀𝑧 ∈ 𝑦 𝜓 → 𝜒)
6		nfcv 2329	. . . . 5 ⊢ Ⅎ𝑦𝑥
7		bdsetindis.nf3	. . . . 5 ⊢ Ⅎ𝑦𝜓
8	6, 7	nfralxy 2525	. . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 𝜓
9		bdsetindis.nf2	. . . 4 ⊢ Ⅎ𝑦𝜑
10	8, 9	nfim 1582	. . 3 ⊢ Ⅎ𝑦(∀𝑧 ∈ 𝑥 𝜓 → 𝜑)
11		raleq 2683	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 𝜓 ↔ ∀𝑧 ∈ 𝑥 𝜓))
12	11	biimprd 158	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑥 𝜓 → ∀𝑧 ∈ 𝑦 𝜓))
13		bdsetindis.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑))
14	13	equcoms 1718	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜒 → 𝜑))
15	12, 14	imim12d 74	. . 3 ⊢ (𝑦 = 𝑥 → ((∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → (∀𝑧 ∈ 𝑥 𝜓 → 𝜑)))
16	5, 10, 15	cbv3 1752	. 2 ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥(∀𝑧 ∈ 𝑥 𝜓 → 𝜑))
17		bdsetindis.1	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓))
18	2, 17	bj-sbime 14821	. . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 → 𝜓)
19	18	ralimi 2550	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → ∀𝑧 ∈ 𝑥 𝜓)
20	19	imim1i 60	. . 3 ⊢ ((∀𝑧 ∈ 𝑥 𝜓 → 𝜑) → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑))
21	20	alimi 1465	. 2 ⊢ (∀𝑥(∀𝑧 ∈ 𝑥 𝜓 → 𝜑) → ∀𝑥(∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑))
22		bdsetindis.bd	. . 3 ⊢ BOUNDED 𝜑
23	22	ax-bdsetind 15016	. 2 ⊢ (∀𝑥(∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)
24	16, 21, 23	3syl 17	1 ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1361  Ⅎwnf 1470  [wsb 1772  ∀wral 2465  BOUNDED wbd 14860

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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-bdsetind 15016

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470

This theorem is referenced by:  bj-inf2vnlem3  15020