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Mirrors > Home > ILE Home > Th. List > nfraldya | GIF version |
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.) |
Ref | Expression |
---|---|
nfraldya.2 | ⊢ Ⅎyφ |
nfraldya.3 | ⊢ (φ → ℲxA) |
nfraldya.4 | ⊢ (φ → Ⅎxψ) |
Ref | Expression |
---|---|
nfraldya | ⊢ (φ → Ⅎx∀y ∈ A ψ) |
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) |
Colors of variables: wff set class |
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 |
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