Theorem ax-9 1470

Description: Derive ax-9 1470 from ax-i9 1469, the modified version for intuitionistic logic. Although ax-9 1470 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1469. (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 1469	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2	1	notnoti 610	. 2 ⊢ ¬ ¬ ∃𝑥 𝑥 = 𝑦
3		alnex 1434	. 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦)
4	2, 3	mtbir 632	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1288   = wceq 1290  ∃wex 1427

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-5 1382  ax-gen 1384  ax-ie2 1429  ax-i9 1469

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296

This theorem is referenced by:  equidqe  1471