Theorem bj-axc14nf 32963

Description: Proof of a version of axc14 2400 using the "non-free" idiom. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axc14nf	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦))

Proof of Theorem bj-axc14nf

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnae 2351	. 2 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2		bj-nfeel2 32962	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥 ∈ 𝑡)
3		elequ2 2044	. 2 ⊢ (𝑡 = 𝑦 → (𝑥 ∈ 𝑡 ↔ 𝑥 ∈ 𝑦))
4	1, 2, 3	bj-dvelimdv1 32960	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521  Ⅎwnf 1748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750

This theorem is referenced by:  bj-axc14  32964