Theorem bj-axc14nf 37337

Description: Proof of a version of axc14 2494 using the "nonfree" 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		bj-nfeel2 37336	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥 ∈ 𝑡)
2		elequ2 2157	. 2 ⊢ (𝑡 = 𝑦 → (𝑥 ∈ 𝑡 ↔ 𝑥 ∈ 𝑦))
3	1, 2	bj-dvelimdv1 37334	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1558  Ⅎwnf 1803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804

This theorem is referenced by:  bj-axc14  37338