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Theorem bj-nfcsym 33011
Description: The class-form not-free predicate defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 4927 with additional axioms; see also nfcv 2793). This could be proved from aecom 2344 and nfcvb 4928 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 2657 instead of equcomd 1992; removing dependency on ax-ext 2631 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2811, eleq2d 2716 (using elequ2 2044), nfcvf 2817, dvelimc 2816, dvelimdc 2815, nfcvf2 2818. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nfcsym (𝑥𝑦𝑦𝑥)

Proof of Theorem bj-nfcsym
StepHypRef Expression
1 sp 2091 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21equcomd 1992 . . 3 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑥)
32drnfc1 2811 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥))
4 nfcvf 2817 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
5 nfcvf2 2818 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
64, 52thd 255 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥))
73, 6pm2.61i 176 1 (𝑥𝑦𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1521  wnfc 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-cleq 2644  df-clel 2647  df-nfc 2782
This theorem is referenced by: (None)
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