Theorem ax9vsep 3991

Description: Derive a weakened version of ax-9 1479, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 3986 and Extensionality ax-ext 2082. In intuitionistic logic a9evsep 3990 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax9vsep	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem ax9vsep

Step	Hyp	Ref	Expression
1		a9evsep 3990	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exalim 1446	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	ax-mp 7	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1297   = wceq 1299  ∃wex 1436

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-ial 1482  ax-ext 2082  ax-sep 3986

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305

This theorem is referenced by: (None)