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Mirrors > Home > MPE Home > Th. List > nfcr | Structured version Visualization version GIF version |
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfcr | ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nfc 2740 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
2 | sp 2041 | . 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sylbi 206 | 1 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nfc 2740 |
This theorem is referenced by: nfcrii 2744 nfcrd 2757 nfnfc 2760 abidnf 3342 csbtt 3510 csbnestgf 3948 bj-nfcrii 32045 |
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