Theorem ax9vsep 4256

Description: Derive a weakened version of ax-9 1584, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4249 and Extensionality ax-ext 2220. In intuitionistic logic a9evsep 4255 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 4255	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exalim 1555	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1400   = wceq 1402  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587  ax-ext 2220  ax-sep 4249

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408

This theorem is used by: (None)