Theorem xfree 29303

Description: A partial converse to 19.9t 2071. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)

Assertion

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

Proof of Theorem xfree

Step	Hyp	Ref	Expression
1		nf5 2116	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		nf6 2117	. 2 |- ( F/ x ph <-> A. x ( E. x ph -> ph ) )
3	1, 2	bitr3i 266	1 |- ( A. x ( ph -> A. x ph ) <-> A. x ( E. x ph -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708

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:  xfree2  29304