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Mirrors > Home > ILE Home > Th. List > nfrexya | GIF version |
Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexxy 2472 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.) |
Ref | Expression |
---|---|
nfralya.1 | ⊢ Ⅎ𝑥𝐴 |
nfralya.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfrexya | ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1442 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfralya.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfralya.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfrexdya 2470 | . 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1340 | 1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1332 Ⅎwnf 1436 Ⅎwnfc 2268 ∃wrex 2417 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 |
This theorem is referenced by: nfiunya 3841 nffrec 6293 nfsup 6879 caucvgsrlemgt1 7603 nfsum1 11125 zsupcllemstep 11638 bezout 11699 |
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