Theorem ax-9 1355

Description: Derive ax-9 1355 from ax-i9 1354, the modified version for intuitionistic logic. Although ax-9 1355 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1354. (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 1354	. . 3 ⊢ ∃x x = y
2	1	notnoti 555	. 2 ⊢ ¬ ¬ ∃x x = y
3		alnex 1320	. 2 ⊢ (∀x ¬ x = y ↔ ¬ ∃x x = y)
4	2, 3	mtbir 577	1 ⊢ ¬ ∀x ¬ x = y

Syntax hints:  ¬ wn 3  ∀wal 1266  ∃wex 1313   = wceq 1324

This theorem is referenced by:  equidqe  1356  equidqeOLD  1357

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in1 527  ax-in2 528  ax-5 1267  ax-gen 1269  ax-ie2 1315  ax-i9 1354

This theorem depends on definitions:  df-bi 108  df-tru 1190  df-fal 1191