Theorem xfree2 29304

Description: A partial converse to 19.9t 2071. (Contributed by Stefan Allan, 21-Dec-2008.)

Assertion

Ref	Expression
xfree2	|- ( A. x ( ph -> A. x ph ) <-> A. x ( -. ph -> A. x -. ph ) )

Proof of Theorem xfree2

Step	Hyp	Ref	Expression
1		xfree 29303	. 2 |- ( A. x ( ph -> A. x ph ) <-> A. x ( E. x ph -> ph ) )
2		eximal 1707	. . 3 |- ( ( E. x ph -> ph ) <-> ( -. ph -> A. x -. ph ) )
3	2	albii 1747	. 2 |- ( A. x ( E. x ph -> ph ) <-> A. x ( -. ph -> A. x -. ph ) )
4	1, 3	bitri 264	1 |- ( A. x ( ph -> A. x ph ) <-> A. x ( -. ph -> A. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

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  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)