Theorem dveel1 2463

Description: Quantifier introduction when one pair of variables is disjoint. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
dveel1	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧))

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveel1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ1 2117	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
2	1	dvelimv 2454	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2374

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791

This theorem is referenced by:  distel  33773