Theorem ax9vsep 4051

Description: Derive a weakened version of ax-9 1511, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4046 and Extensionality ax-ext 2121. In intuitionistic logic a9evsep 4050 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 4050	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		exalim 1478	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1329   = wceq 1331  ∃wex 1468

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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-ext 2121  ax-sep 4046

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337

This theorem is referenced by: (None)