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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfcsym | Structured version Visualization version GIF version | ||
| Description: The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5374 with additional axioms; see also nfcv 2904). This could be proved from aecom 2431 and nfcvb 5375 but the latter requires a domain with at least two objects (hence uses extra axioms). (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid use of eqcomd 2742 instead of equcomd 2017; removing dependency on ax-ext 2707 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2924, eleq2d 2826 (using elequ2 2122), nfcvf 2931, dvelimc 2930, dvelimdc 2929, nfcvf2 2932. (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-nfcsym | ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sp 2182 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
| 2 | 1 | equcomd 2017 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥) | 
| 3 | 2 | drnfc1 2924 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) | 
| 4 | nfcvf 2931 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
| 5 | nfcvf2 2932 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) | |
| 6 | 4, 5 | 2thd 265 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) | 
| 7 | 3, 6 | pm2.61i 182 | 1 ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1537 Ⅎwnfc 2889 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-cleq 2728 df-nfc 2891 | 
| This theorem is referenced by: (None) | 
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