![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 2427 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 2739 instead of equcomd 2023; removing dependency on ax-ext 2704 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2923, eleq2d 2820 (using elequ2 2122), nfcvf 2933, dvelimc 2932, dvelimdc 2931, nfcvf2 2934. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nfcsym | ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2177 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | 1 | equcomd 2023 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥) |
3 | 2 | drnfc1 2923 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) |
4 | nfcvf 2933 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
5 | nfcvf2 2934 | . . 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 205 ∀wal 1540 Ⅎwnfc 2884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2372 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-cleq 2725 df-nfc 2886 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |