Theorem bj-nfcsym 35157

Description: The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5313 with additional axioms; see also nfcv 2905). This could be proved from aecom 2426 and nfcvb 5314 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 2743 instead of equcomd 2021; removing dependency on ax-ext 2708 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2924, eleq2d 2823 (using elequ2 2120), nfcvf 2934, dvelimc 2933, dvelimdc 2932, nfcvf2 2935. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nfcsym	⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)

Proof of Theorem bj-nfcsym

Step	Hyp	Ref	Expression
1		sp 2175	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	equcomd 2021	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥)
3	2	drnfc1 2924	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥))
4		nfcvf 2934	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
5		nfcvf2 2935	. . 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 1538  Ⅎwnfc 2885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2371  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-cleq 2729  df-nfc 2887

This theorem is referenced by: (None)