Theorem ax9dgen 1716

Description: Tarski's system uses the weaker ax9v 1655 instead of the bundled ax-9 1654, so here we show that the degenerate case of ax-9 1654 can be derived. (Contributed by NM, 23-Apr-2017.)

Assertion

Ref	Expression
ax9dgen	⊢ ¬ ∀x ¬ x = x

Proof of Theorem ax9dgen

Step	Hyp	Ref	Expression
1		equid 1676	. 2 ⊢ x = x
2	1	notnoti 115	. . 3 ⊢ ¬ ¬ x = x
3	2	spfalw 1672	. 2 ⊢ (∀x ¬ x = x → ¬ x = x)
4	1, 3	mt2 170	1 ⊢ ¬ ∀x ¬ x = x

Syntax hints:  ¬ wn 3  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)