Theorem ax6blem 1540

Description: If x is not free in ph, it is not free in -. ph. This theorem doesn't use ax6b 1541 compared to hbnt 1543. (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 1484	. . 3 |- ( E. x ph -> ph )
4	3	con3i 562	. 2 |- ( -. ph -> -. E. x ph )
5		alnex 1388	. 2 |- ( A. x -. ph <-> -. E. x ph )
6	4, 5	sylibr 137	1 |- ( -. ph -> A. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by:  ax6b  1541