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Mirrors > Home > ILE Home > Th. List > nfrexdxy | GIF version |
Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2353 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) |
Ref | Expression |
---|---|
nfraldxy.2 | ⊢ Ⅎyφ |
nfraldxy.3 | ⊢ (φ → ℲxA) |
nfraldxy.4 | ⊢ (φ → Ⅎxψ) |
Ref | Expression |
---|---|
nfrexdxy | ⊢ (φ → Ⅎx∃y ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2306 | . 2 ⊢ (∃y ∈ A ψ ↔ ∃y(y ∈ A ∧ ψ)) | |
2 | nfraldxy.2 | . . 3 ⊢ Ⅎyφ | |
3 | nfcv 2175 | . . . . . 6 ⊢ Ⅎxy | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (φ → Ⅎxy) |
5 | nfraldxy.3 | . . . . 5 ⊢ (φ → ℲxA) | |
6 | 4, 5 | nfeld 2190 | . . . 4 ⊢ (φ → Ⅎx y ∈ A) |
7 | nfraldxy.4 | . . . 4 ⊢ (φ → Ⅎxψ) | |
8 | 6, 7 | nfand 1457 | . . 3 ⊢ (φ → Ⅎx(y ∈ A ∧ ψ)) |
9 | 2, 8 | nfexd 1641 | . 2 ⊢ (φ → Ⅎx∃y(y ∈ A ∧ ψ)) |
10 | 1, 9 | nfxfrd 1361 | 1 ⊢ (φ → Ⅎx∃y ∈ A ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 Ⅎwnf 1346 ∃wex 1378 ∈ wcel 1390 Ⅎwnfc 2162 ∃wrex 2301 |
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-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 |
This theorem is referenced by: nfrexdya 2353 nfrexxy 2355 nfunid 3578 strcollnft 9444 |
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