Theorem ax-9 1421

Description: Derive ax-9 1421 from ax-i9 1420, the modified version for intuitionistic logic. Although ax-9 1421 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1420. (Contributed by NM, 3-Feb-2015.)

Assertion

Ref	Expression
ax-9	⊢ ¬ ∀x ¬ x = y

Proof of Theorem ax-9

Step	Hyp	Ref	Expression
1		ax-i9 1420	. . 3 ⊢ ∃x x = y
2	1	notnoti 573	. 2 ⊢ ¬ ¬ ∃x x = y
3		alnex 1385	. 2 ⊢ (∀x ¬ x = y ↔ ¬ ∃x x = y)
4	2, 3	mtbir 595	1 ⊢ ¬ ∀x ¬ x = y

Syntax hints:  ¬ wn 3  ∀wal 1240   = wceq 1242  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie2 1380  ax-i9 1420

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248

This theorem is referenced by:  equidqe  1422