Theorem bj-nfeel2 32030

Description: Non-freeness in an equality. (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 1830	. 2 ⊢ Ⅎ𝑥 𝑡 ∈ 𝑧
2		eleq1 2676	. 2 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
3	1, 2	bj-dvelimv 32029	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606

This theorem is referenced by:  bj-axc14nf  32031