Theorem ax-9 1512

Description: Derive ax-9 1512 from ax-i9 1511, the modified version for intuitionistic logic. Although ax-9 1512 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1511. (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 1511	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2	1	notnoti 635	. 2 ⊢ ¬ ¬ ∃𝑥 𝑥 = 𝑦
3		alnex 1476	. 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦)
4	2, 3	mtbir 661	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1330   = wceq 1332  ∃wex 1469

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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie2 1471  ax-i9 1511

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338

This theorem is referenced by:  equidqe  1513