Theorem bj-nfcsym 37907

Description: The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5337 with additional axioms; see also nfcv 2922). This could be proved from aecom 2456 and nfcvb 5338 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 2766 instead of equcomd 2051; removing dependency on ax-ext 2732 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2941, eleq2d 2846 (using elequ2 2159), nfcvf 2948, dvelimc 2947, dvelimdc 2946, nfcvf2 2949. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nfcsym	⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)

Proof of Theorem bj-nfcsym

Step	Hyp	Ref	Expression
1		sp 2219	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	equcomd 2051	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥)
3	2	drnfc1 2941	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥))
4		nfcvf 2948	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
5		nfcvf2 2949	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
6	4, 5	2thd 268	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥))
7	3, 6	pm2.61i 184	1 ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1567  Ⅎwnfc 2907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-cleq 2752  df-nfc 2909

This theorem is used by: (None)