Theorem dveel2 2500

Description: Quantifier introduction when one pair of variables is disjoint. Usage of this theorem is discouraged because it depends on ax-13 2410. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
dveel2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2164	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
2	1	dvelimv 2490	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811

This theorem is referenced by:  axc14  2501