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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 2505 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 1453 | . . 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 2498 | . 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1351 | 1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1343 Ⅎwnf 1447 Ⅎwnfc 2293 ∃wrex 2443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 |
This theorem is referenced by: r19.12 2570 sbcrext 3023 nfuni 3789 nfiunxy 3886 rexxpf 4745 abrexex2g 6080 abrexex2 6084 nfrecs 6266 fimaxre2 11154 nfsum 11284 nfcprod1 11481 nfcprod 11482 bezoutlemmain 11916 ctiunctlemfo 12309 bj-findis 13696 strcollnfALT 13703 |
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