Theorem setindis 15977

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 2349	. . . . 5 ⊢ Ⅎ𝑥𝑦
2		setindis.nf0	. . . . 5 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfralxy 2545	. . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 𝜓
4		setindis.nf1	. . . 4 ⊢ Ⅎ𝑥𝜒
5	3, 4	nfim 1596	. . 3 ⊢ Ⅎ𝑥(∀𝑧 ∈ 𝑦 𝜓 → 𝜒)
6		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑦𝑥
7		setindis.nf3	. . . . 5 ⊢ Ⅎ𝑦𝜓
8	6, 7	nfralxy 2545	. . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 𝜓
9		setindis.nf2	. . . 4 ⊢ Ⅎ𝑦𝜑
10	8, 9	nfim 1596	. . 3 ⊢ Ⅎ𝑦(∀𝑧 ∈ 𝑥 𝜓 → 𝜑)
11		raleq 2703	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 𝜓 ↔ ∀𝑧 ∈ 𝑥 𝜓))
12	11	biimprd 158	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑥 𝜓 → ∀𝑧 ∈ 𝑦 𝜓))
13		setindis.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑))
14	13	equcoms 1732	. . . 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 1766	. 2 ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥(∀𝑧 ∈ 𝑥 𝜓 → 𝜑))
17		setindis.1	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓))
18	2, 17	bj-sbime 15783	. . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 → 𝜓)
19	18	ralimi 2570	. . . 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 1479	. 2 ⊢ (∀𝑥(∀𝑧 ∈ 𝑥 𝜓 → 𝜑) → ∀𝑥(∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑))
22		ax-setind 4589	. 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 1371  Ⅎwnf 1484  [wsb 1786  ∀wral 2485

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-setind 4589

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490

This theorem is referenced by:  bj-inf2vnlem4  15983  bj-findis  15989