Theorem bdsetindis 9429

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 2175	. . . . 5 ⊢ Ⅎxy
2		bdsetindis.nf0	. . . . 5 ⊢ Ⅎxψ
3	1, 2	nfralxy 2354	. . . 4 ⊢ Ⅎx∀z ∈ y ψ
4		bdsetindis.nf1	. . . 4 ⊢ Ⅎxχ
5	3, 4	nfim 1461	. . 3 ⊢ Ⅎx(∀z ∈ y ψ → χ)
6		nfcv 2175	. . . . 5 ⊢ Ⅎyx
7		bdsetindis.nf3	. . . . 5 ⊢ Ⅎyψ
8	6, 7	nfralxy 2354	. . . 4 ⊢ Ⅎy∀z ∈ x ψ
9		bdsetindis.nf2	. . . 4 ⊢ Ⅎyφ
10	8, 9	nfim 1461	. . 3 ⊢ Ⅎy(∀z ∈ x ψ → φ)
11		raleq 2499	. . . . 5 ⊢ (y = x → (∀z ∈ y ψ ↔ ∀z ∈ x ψ))
12	11	biimprd 147	. . . 4 ⊢ (y = x → (∀z ∈ x ψ → ∀z ∈ y ψ))
13		bdsetindis.2	. . . . 5 ⊢ (x = y → (χ → φ))
14	13	equcoms 1591	. . . 4 ⊢ (y = x → (χ → φ))
15	12, 14	imim12d 68	. . 3 ⊢ (y = x → ((∀z ∈ y ψ → χ) → (∀z ∈ x ψ → φ)))
16	5, 10, 15	cbv3 1627	. 2 ⊢ (∀y(∀z ∈ y ψ → χ) → ∀x(∀z ∈ x ψ → φ))
17		bdsetindis.1	. . . . . 6 ⊢ (x = z → (φ → ψ))
18	2, 17	bj-sbime 9248	. . . . 5 ⊢ ([z / x]φ → ψ)
19	18	ralimi 2378	. . . 4 ⊢ (∀z ∈ x [z / x]φ → ∀z ∈ x ψ)
20	19	imim1i 54	. . 3 ⊢ ((∀z ∈ x ψ → φ) → (∀z ∈ x [z / x]φ → φ))
21	20	alimi 1341	. 2 ⊢ (∀x(∀z ∈ x ψ → φ) → ∀x(∀z ∈ x [z / x]φ → φ))
22		bdsetindis.bd	. . 3 ⊢ BOUNDED φ
23	22	ax-bdsetind 9428	. 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 1240  Ⅎwnf 1346  [wsb 1642  ∀wral 2300  BOUNDED wbd 9267

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bdsetind 9428

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305

This theorem is referenced by:  bj-inf2vnlem3  9432