Theorem setindis 14002

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

Hypotheses

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

Assertion

Ref	Expression
setindis	⊢ (∀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 setindis

Step	Hyp	Ref	Expression
1		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑥𝑦
2		setindis.nf0	. . . . 5 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfralxy 2508	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 𝜓
4		setindis.nf1	. . . 4 ⊢ Ⅎ𝑥𝜒
5	3, 4	nfim 1565	. . 3 ⊢ Ⅎ𝑥(∀𝑧 ∈ 𝑦 𝜓 → 𝜒)
6		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑦𝑥
7		setindis.nf3	. . . . 5 ⊢ Ⅎ𝑦𝜓
8	6, 7	nfralxy 2508	. . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 𝜓
9		setindis.nf2	. . . 4 ⊢ Ⅎ𝑦𝜑
10	8, 9	nfim 1565	. . 3 ⊢ Ⅎ𝑦(∀𝑧 ∈ 𝑥 𝜓 → 𝜑)
11		raleq 2665	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 𝜓 ↔ ∀𝑧 ∈ 𝑥 𝜓))
12	11	biimprd 157	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑥 𝜓 → ∀𝑧 ∈ 𝑦 𝜓))
13		setindis.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑))
14	13	equcoms 1701	. . . 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 1735	. 2 ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥(∀𝑧 ∈ 𝑥 𝜓 → 𝜑))
17		setindis.1	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓))
18	2, 17	bj-sbime 13808	. . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 → 𝜓)
19	18	ralimi 2533	. . . 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 1448	. 2 ⊢ (∀𝑥(∀𝑧 ∈ 𝑥 𝜓 → 𝜑) → ∀𝑥(∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑))
22		ax-setind 4521	. 2 ⊢ (∀𝑥(∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)
23	16, 21, 22	3syl 17	1 ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1346  Ⅎwnf 1453  [wsb 1755  ∀wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453

This theorem is referenced by:  bj-inf2vnlem4  14008  bj-findis  14014