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 32079
 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.)
Assertion
Ref Expression
bj-nfcsym (𝑥𝑦𝑦𝑥)

Proof of Theorem bj-nfcsym
StepHypRef Expression
1 sp 2041 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21equcomd 1933 . . 3 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑥)
32drnfc1 2768 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥))
4 nfcvf 2774 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
5 nfcvf2 2775 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
64, 52thd 254 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥))
73, 6pm2.61i 175 1 (𝑥𝑦𝑦𝑥)
 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)
 Copyright terms: Public domain W3C validator