Theorem bj-nfeel2 34180

Description: Nonfreeness in a membership statement. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nfeel2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)

Distinct variable group:   𝑥,𝑧

Proof of Theorem bj-nfeel2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑥 𝑡 ∈ 𝑧
2		elequ1 2121	. 2 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
3	1, 2	bj-dvelimv 34179	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  Ⅎwnf 1784

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785

This theorem is referenced by:  bj-axc14nf  34181