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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 4823 with additional axioms; see also nfcv 2751). This could be proved from aecom 2299 and nfcvb 4824 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 2616 instead of equcomd 1933; removing dependency on ax-ext 2590 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2768, eleq2d 2673 (using elequ2 1991), nfcvf 2774, dvelimc 2773, dvelimdc 2772, nfcvf2 2775. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nfcsym | ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2041 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | 1 | equcomd 1933 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥) |
3 | 2 | drnfc1 2768 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) |
4 | nfcvf 2774 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
5 | nfcvf2 2775 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) | |
6 | 4, 5 | 2thd 254 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) |
7 | 3, 6 | pm2.61i 175 | 1 ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wal 1473 Ⅎwnfc 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-cleq 2603 df-clel 2606 df-nfc 2740 |
This theorem is referenced by: (None) |
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