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Theorem bj-nfcsym 36882
Description: The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5381 with additional axioms; see also nfcv 2903). This could be proved from aecom 2430 and nfcvb 5382 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 2741 instead of equcomd 2016; removing dependency on ax-ext 2706 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2923, eleq2d 2825 (using elequ2 2121), nfcvf 2930, dvelimc 2929, dvelimdc 2928, nfcvf2 2931. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nfcsym (𝑥𝑦𝑦𝑥)

Proof of Theorem bj-nfcsym
StepHypRef Expression
1 sp 2181 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21equcomd 2016 . . 3 (∀𝑥 𝑥 = 𝑦𝑦 = 𝑥)
32drnfc1 2923 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥))
4 nfcvf 2930 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
5 nfcvf2 2931 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
64, 52thd 265 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥𝑦𝑦𝑥))
73, 6pm2.61i 182 1 (𝑥𝑦𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wal 1535  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-cleq 2727  df-nfc 2890
This theorem is referenced by: (None)
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