Theorem ax6dgen 2005

Description: Tarski's system uses the weaker ax6v 1889 instead of the bundled ax-6 1888, so here we show that the degenerate case of ax-6 1888 can be derived. Even though ax-6 1888 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1888 is ax6v 1889. We later rederive from ax6v 1889 the bundled form as ax6 2251 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)

Assertion

Ref	Expression
ax6dgen	|- -. A. x -. x = x

Proof of Theorem ax6dgen

Step	Hyp	Ref	Expression
1		equid 1939	. 2 |- x = x
2	1	notnoti 137	. . 3 |- -. -. x = x
3	2	spfalw 1929	. 2 |- ( A. x -. x = x -> -. x = x )
4	1, 3	mt2 191	1 |- -. A. x -. x = x

Syntax hints:   -. wn 3   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)