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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.) |
Ref | Expression |
---|---|
nfcri.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfcrii | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcri.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcriv 2925 | . . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
3 | 2 | nf5ri 2179 | . 2 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) |
4 | 3 | hblem 2891 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1599 ∈ wcel 2107 Ⅎwnfc 2919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clel 2774 df-nfc 2921 |
This theorem is referenced by: nfcri 2929 cleqfOLD 2964 abeq2fOLD 2967 bnj1230 31472 bnj1000 31610 bnj1204 31679 bnj1307 31690 bnj1311 31691 bnj1398 31701 bnj1466 31720 bnj1467 31721 bnj1523 31738 |
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