Theorem ax9vsep 4138

Description: Derive a weakened version of ax-9 1541, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4133 and Extensionality ax-ext 2169. In intuitionistic logic a9evsep 4137 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 4137	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exalim 1512	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1361   = wceq 1363  ∃wex 1502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544  ax-ext 2169  ax-sep 4133

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369

This theorem is referenced by: (None)