Theorem nfraldya 2352

Description: Not-free for restricted universal quantification where y and A are distinct. See nfraldxy 2350 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

Ref	Expression
nfraldya.2	⊢ Ⅎyφ
nfraldya.3	⊢ (φ → ℲxA)
nfraldya.4	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfraldya	⊢ (φ → Ⅎx∀y ∈ A ψ)

Distinct variable group:   y,A

Allowed substitution hints:   φ(x,y)   ψ(x,y)   A(x)

Proof of Theorem nfraldya

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 2305	. 2 ⊢ (∀y ∈ A ψ ↔ ∀y(y ∈ A → ψ))
2		sbim 1824	. . . . . 6 ⊢ ([z / y](y ∈ A → ψ) ↔ ([z / y]y ∈ A → [z / y]ψ))
3		clelsb3 2139	. . . . . . 7 ⊢ ([z / y]y ∈ A ↔ z ∈ A)
4	3	imbi1i 227	. . . . . 6 ⊢ (([z / y]y ∈ A → [z / y]ψ) ↔ (z ∈ A → [z / y]ψ))
5	2, 4	bitri 173	. . . . 5 ⊢ ([z / y](y ∈ A → ψ) ↔ (z ∈ A → [z / y]ψ))
6	5	albii 1356	. . . 4 ⊢ (∀z[z / y](y ∈ A → ψ) ↔ ∀z(z ∈ A → [z / y]ψ))
7		nfv 1418	. . . . 5 ⊢ Ⅎz(y ∈ A → ψ)
8	7	sb8 1733	. . . 4 ⊢ (∀y(y ∈ A → ψ) ↔ ∀z[z / y](y ∈ A → ψ))
9		df-ral 2305	. . . 4 ⊢ (∀z ∈ A [z / y]ψ ↔ ∀z(z ∈ A → [z / y]ψ))
10	6, 8, 9	3bitr4i 201	. . 3 ⊢ (∀y(y ∈ A → ψ) ↔ ∀z ∈ A [z / y]ψ)
11		nfv 1418	. . . 4 ⊢ Ⅎzφ
12		nfraldya.3	. . . 4 ⊢ (φ → ℲxA)
13		nfraldya.2	. . . . 5 ⊢ Ⅎyφ
14		nfraldya.4	. . . . 5 ⊢ (φ → Ⅎxψ)
15	13, 14	nfsbd 1848	. . . 4 ⊢ (φ → Ⅎx[z / y]ψ)
16	11, 12, 15	nfraldxy 2350	. . 3 ⊢ (φ → Ⅎx∀z ∈ A [z / y]ψ)
17	10, 16	nfxfrd 1361	. 2 ⊢ (φ → Ⅎx∀y(y ∈ A → ψ))
18	1, 17	nfxfrd 1361	1 ⊢ (φ → Ⅎx∀y ∈ A ψ)

Syntax hints:   → wi 4  ∀wal 1240  Ⅎwnf 1346   ∈ wcel 1390  [wsb 1642  Ⅎwnfc 2162  ∀wral 2300

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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

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

This theorem is referenced by:  nfralya  2356