Theorem ax9vsep 4059

Description: Derive a weakened version of ax-9 1512, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4054 and Extensionality ax-ext 2122. In intuitionistic logic a9evsep 4058 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 4058	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exalim 1479	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1330   = wceq 1332  ∃wex 1469

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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515  ax-ext 2122  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338

This theorem is referenced by: (None)