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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfcsym | Structured version Visualization version GIF version |
Description: The class-form not-free predicate defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 4927 with additional axioms; see also nfcv 2793). This could be proved from aecom 2344 and nfcvb 4928 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 2657 instead of equcomd 1992; removing dependency on ax-ext 2631 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2811, eleq2d 2716 (using elequ2 2044), nfcvf 2817, dvelimc 2816, dvelimdc 2815, nfcvf2 2818. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nfcsym | ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2091 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | 1 | equcomd 1992 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥) |
3 | 2 | drnfc1 2811 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) |
4 | nfcvf 2817 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
5 | nfcvf2 2818 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) | |
6 | 4, 5 | 2thd 255 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) |
7 | 3, 6 | pm2.61i 176 | 1 ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∀wal 1521 Ⅎwnfc 2780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-cleq 2644 df-clel 2647 df-nfc 2782 |
This theorem is referenced by: (None) |
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