Theorem dveeq2 2378

Description: Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2154. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
dveeq2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2

Step	Hyp	Ref	Expression
1		nfeqf2 2377	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
2	1	nf5rd 2189	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787

This theorem is referenced by:  axc15  2422