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Mirrors > Home > MPE Home > Th. List > df-nfc | Structured version Visualization version GIF version |
Description: Define the not-free predicate for classes. This is read "𝑥 is not free in 𝐴". Not-free means that the value of 𝑥 cannot affect the value of 𝐴, e.g., any occurrence of 𝑥 in 𝐴 is effectively bound by a "for all" or something that expands to one (such as "there exists"). It is defined in terms of the not-free predicate df-nf 1776 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
df-nfc | ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . 3 setvar 𝑥 | |
2 | cA | . . 3 class 𝐴 | |
3 | 1, 2 | wnfc 2961 | . 2 wff Ⅎ𝑥𝐴 |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 4 | cv 1527 | . . . . 5 class 𝑦 |
6 | 5, 2 | wcel 2105 | . . . 4 wff 𝑦 ∈ 𝐴 |
7 | 6, 1 | wnf 1775 | . . 3 wff Ⅎ𝑥 𝑦 ∈ 𝐴 |
8 | 7, 4 | wal 1526 | . 2 wff ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 |
9 | 3, 8 | wb 207 | 1 wff (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
This definition is referenced by: nfci 2964 nfcr 2966 nfcd 2968 nfceqdf 2972 nfceqi 2973 nfnfc1 2980 nfnfc 2990 drnfc1 2997 drnfc1OLD 2998 drnfc2 2999 dfnfc2 4850 nfnid 5268 nfriotadw 7111 bj-nfcf 34140 |
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