Theorem ax-9 1464

Description: Derive ax-9 1464 from ax-i9 1463, the modified version for intuitionistic logic. Although ax-9 1464 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1463. (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 1463	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2	1	notnoti 606	. 2 ⊢ ¬ ¬ ∃𝑥 𝑥 = 𝑦
3		alnex 1428	. 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦)
4	2, 3	mtbir 628	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1282   = wceq 1284  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-i9 1463

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  equidqe  1465