Theorem ax9vsep 4112

Description: Derive a weakened version of ax-9 1524, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4107 and Extensionality ax-ext 2152. In intuitionistic logic a9evsep 4111 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 4111	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exalim 1495	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1346   = wceq 1348  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by: (None)