Theorem ax-9 1541

Description: Derive ax-9 1541 from ax-i9 1540, the modified version for intuitionistic logic. Although ax-9 1541 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1540. (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 1540	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
2	1	notnoti 646	. 2 ⊢ ¬ ¬ ∃𝑥 𝑥 = 𝑦
3		alnex 1509	. 2 ⊢ (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦)
4	2, 3	mtbir 672	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1361   = wceq 1363  ∃wex 1502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-ie2 1504  ax-i9 1540

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369

This theorem is referenced by:  equidqe  1542