Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfcsym | Structured version Visualization version Unicode version |
Description: The class-form not-free predicate defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 4897 with additional axioms; see also nfcv 2764). This could be proved from aecom 2311 and nfcvb 4898 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 2628 instead of equcomd 1946; removing dependency on ax-ext 2602 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2782, eleq2d 2687 (using elequ2 2004), nfcvf 2788, dvelimc 2787, dvelimdc 2786, nfcvf2 2789. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nfcsym |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2053 | . . . 4 | |
2 | 1 | equcomd 1946 | . . 3 |
3 | 2 | drnfc1 2782 | . 2 |
4 | nfcvf 2788 | . . 3 | |
5 | nfcvf2 2789 | . . 3 | |
6 | 4, 5 | 2thd 255 | . 2 |
7 | 3, 6 | pm2.61i 176 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 196 wal 1481 wnfc 2751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-cleq 2615 df-clel 2618 df-nfc 2753 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |