Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-nfcsym Structured version   Visualization version   GIF version

Theorem bj-nfcsym 31911
Description: The class-form not-free predicate defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 4722 with additional axioms; see also nfcv 2655). This could be proved from aecom 2203 and nfcvb 4723 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 2520 instead of equcomd 1896; removing dependency on ax-ext 2494 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2672, eleq2d 2577 (using elequ2 1952), nfcvf 2678, dvelimc 2677, dvelimdc 2676, nfcvf2 2679. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nfcsym (𝑥𝑦𝑦𝑥)

Proof of Theorem bj-nfcsym
StepHypRef Expression
1 sp 1990 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21equcomd 1896 . . 3 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑥)
32drnfc1 2672 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥))
4 nfcvf 2678 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
5 nfcvf2 2679 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
64, 52thd 253 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥))
73, 6pm2.61i 174 1 (𝑥𝑦𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wal 1472  wnfc 2642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-cleq 2507  df-clel 2510  df-nfc 2644
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator