Theorem bj-nfcsym 35865

Description: The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5373 with additional axioms; see also nfcv 2903). This could be proved from aecom 2426 and nfcvb 5374 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 2738 instead of equcomd 2022; removing dependency on ax-ext 2703 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2922, eleq2d 2819 (using elequ2 2121), nfcvf 2932, dvelimc 2931, dvelimdc 2930, nfcvf2 2933. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nfcsym	⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)

Proof of Theorem bj-nfcsym

Step	Hyp	Ref	Expression
1		sp 2176	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	equcomd 2022	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥)
3	2	drnfc1 2922	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥))
4		nfcvf 2932	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
5		nfcvf2 2933	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
6	4, 5	2thd 264	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥))
7	3, 6	pm2.61i 182	1 ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1539  Ⅎwnfc 2883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-cleq 2724  df-nfc 2885

This theorem is referenced by: (None)