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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 2361 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 1355 | . . 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 2359 | . 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑) |
7 | 6 | trud 1252 | 1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1244 Ⅎwnf 1349 Ⅎwnfc 2165 ∃wrex 2307 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 |
This theorem is referenced by: nfiunya 3685 nffrec 5982 caucvgsrlemgt1 6879 nfsum1 9875 nfsum 9876 |
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