Theorem bj-nfcsym 37147

Description: The nonfreeness quantifier for classes defines a symmetric binary relation on var metavariables (irreflexivity is proved by nfnid 5322 with additional axioms; see also nfcv 2899). This could be proved from aecom 2432 and nfcvb 5323 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 2709 is possible: prove weak versions (i.e. replace classes with setvars) of drnfc1 2919, eleq2d 2823 (using elequ2 2129), nfcvf 2926, dvelimc 2925, dvelimdc 2924, nfcvf2 2927. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nfcsym	⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)

Proof of Theorem bj-nfcsym

Step	Hyp	Ref	Expression
1		sp 2191	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	equcomd 2021	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑦 = 𝑥)
3	2	drnfc1 2919	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥))
4		nfcvf 2926	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
5		nfcvf2 2927	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
6	4, 5	2thd 265	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥))
7	3, 6	pm2.61i 182	1 ⊢ (Ⅎ𝑥𝑦 ↔ Ⅎ𝑦𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1540  Ⅎwnfc 2884

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 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-cleq 2729  df-nfc 2886

This theorem is referenced by: (None)