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Mirrors > Home > ILE Home > Th. List > nfcrii | GIF version |
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfcri.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfcrii | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcri.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcr 2273 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
4 | 3 | nfri 1499 | . 2 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) |
5 | 4 | hblem 2247 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2268 |
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-nf 1437 df-sb 1736 df-cleq 2132 df-clel 2135 df-nfc 2270 |
This theorem is referenced by: nfcri 2275 |
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