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Mirrors > Home > ILE Home > Th. List > nfrexxy | GIF version |
Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2357 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) |
Ref | Expression |
---|---|
nfralxy.1 | ⊢ ℲxA |
nfralxy.2 | ⊢ Ⅎxφ |
Ref | Expression |
---|---|
nfrexxy | ⊢ Ⅎx∃y ∈ A φ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1352 | . . 3 ⊢ Ⅎy ⊤ | |
2 | nfralxy.1 | . . . 4 ⊢ ℲxA | |
3 | 2 | a1i 9 | . . 3 ⊢ ( ⊤ → ℲxA) |
4 | nfralxy.2 | . . . 4 ⊢ Ⅎxφ | |
5 | 4 | a1i 9 | . . 3 ⊢ ( ⊤ → Ⅎxφ) |
6 | 1, 3, 5 | nfrexdxy 2351 | . 2 ⊢ ( ⊤ → Ⅎx∃y ∈ A φ) |
7 | 6 | trud 1251 | 1 ⊢ Ⅎx∃y ∈ A φ |
Colors of variables: wff set class |
Syntax hints: ⊤ wtru 1243 Ⅎwnf 1346 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-tru 1245 df-nf 1347 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 |
This theorem is referenced by: r19.12 2416 sbcrext 2829 sbcrexg 2831 nfuni 3577 nfiunxy 3674 rexxpf 4426 abrexex2g 5689 abrexex2 5693 nfrecs 5863 bj-findis 9439 strcollnfALT 9446 |
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