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Mirrors > Home > MPE Home > Th. List > nfcrii | Structured version Visualization version GIF version |
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2129, ax-11 2146. (Revised by Gino Giotto, 23-May-2024.) |
Ref | Expression |
---|---|
nfcrii.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfcrii | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcrii.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcri 2882 | . 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
3 | 2 | nf5ri 2180 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∈ wcel 2098 Ⅎwnfc 2875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 df-clel 2802 df-nfc 2877 |
This theorem is referenced by: nfcriOLDOLDOLD 2889 bnj1230 34302 bnj1000 34441 bnj1204 34512 bnj1307 34523 bnj1311 34524 bnj1398 34534 bnj1466 34553 bnj1467 34554 bnj1523 34571 |
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