Theorem ax6blem 1580

Description: If x is not free in ph, it is not free in -. ph. This theorem doesn't use ax6b 1581 compared to hbnt 1583. (Contributed by GD, 27-Jan-2018.)

Hypothesis

Ref	Expression
ax6blem.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
ax6blem	|- ( -. ph -> A. x -. ph )

Proof of Theorem ax6blem

Step	Hyp	Ref	Expression
1		ax6blem.1	. . . 4 |- ( ph -> A. x ph )
2		id 19	. . . 4 |- ( ph -> ph )
3	1, 2	exlimih 1524	. . 3 |- ( E. x ph -> ph )
4	3	con3i 594	. 2 |- ( -. ph -> -. E. x ph )
5		alnex 1428	. 2 |- ( A. x -. ph <-> -. E. x ph )
6	4, 5	sylibr 132	1 |- ( -. ph -> A. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie2 1423

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  ax6b  1581