Theorem dveeq2-o 39052

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2380 using ax-c15 39008. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2-o

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1911	. 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2		ax-5 1911	. 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3		equequ2 2027	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
4	1, 2, 3	dvelimf-o 39048	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

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 1968  ax-7 2009  ax-10 2146  ax-11 2162  ax-12 2182  ax-13 2374  ax-c5 39002  ax-c4 39003  ax-c7 39004  ax-c10 39005  ax-c11 39006  ax-c9 39009

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785

This theorem is referenced by:  ax12eq  39060  ax12el  39061  ax12inda  39067  ax12v2-o  39068