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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 2321 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
4 | 3 | nfri 1529 | . 2 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) |
5 | 4 | hblem 2295 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1361 Ⅎwnf 1470 ∈ wcel 2158 Ⅎwnfc 2316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-cleq 2180 df-clel 2183 df-nfc 2318 |
This theorem is referenced by: nfcri 2323 |
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