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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 5381 with additional axioms; see also nfcv 2903). This could be proved from aecom 2430 and nfcvb 5382 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 2741 instead of equcomd 2016; removing dependency on ax-ext 2706 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2923, eleq2d 2825 (using elequ2 2121), nfcvf 2930, dvelimc 2929, dvelimdc 2928, nfcvf2 2931. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nfcsym | ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2181 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | 1 | equcomd 2016 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥) |
3 | 2 | drnfc1 2923 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) |
4 | nfcvf 2930 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
5 | nfcvf2 2931 | . . 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 1535 Ⅎwnfc 2888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-cleq 2727 df-nfc 2890 |
This theorem is referenced by: (None) |
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