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Mirrors > Home > ILE Home > Th. List > nfralya | GIF version |
Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfralxy 2408 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 |
---|---|
nfralya | ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1396 | . . 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 | nfraldya 2406 | . 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑) |
7 | 6 | trud 1294 | 1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1286 Ⅎwnf 1390 Ⅎwnfc 2210 ∀wral 2353 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 |
This theorem is referenced by: nfiinya 3733 nfsup 6593 caucvgsrlemgt1 7242 supinfneg 8977 infsupneg 8978 |
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