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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 5267 with additional axioms; see also nfcv 2975). This could be proved from aecom 2443 and nfcvb 5268 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 2825 instead of equcomd 2020; removing dependency on ax-ext 2791 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2995, eleq2d 2896 (using elequ2 2123), nfcvf 3005, dvelimc 3004, dvelimdc 3003, nfcvf2 3006. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nfcsym | ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2175 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | 1 | equcomd 2020 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥) |
3 | 2 | drnfc1 2995 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) |
4 | nfcvf 3005 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
5 | nfcvf2 3006 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) | |
6 | 4, 5 | 2thd 267 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)) |
7 | 3, 6 | pm2.61i 184 | 1 ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∀wal 1529 Ⅎwnfc 2959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-13 2384 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-cleq 2812 df-clel 2891 df-nfc 2961 |
This theorem is referenced by: (None) |
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