Theorem nfbiit 1777

Description: Equivalence theorem for the non-freeness predicate. Closed form of nfbii 1778. (Contributed by BJ, 6-May-2019.)

Assertion

Ref	Expression
nfbiit	|- ( A. x ( ph <-> ps ) -> ( F/ x ph <-> F/ x ps ) )

Proof of Theorem nfbiit

Step	Hyp	Ref	Expression
1		exbi 1773	. . 3 |- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) )
2		albi 1746	. . 3 |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) )
3	1, 2	imbi12d 334	. 2 |- ( A. x ( ph <-> ps ) -> ( ( E. x ph -> A. x ph ) <-> ( E. x ps -> A. x ps ) ) )
4		df-nf 1710	. 2 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
5		df-nf 1710	. 2 |- ( F/ x ps <-> ( E. x ps -> A. x ps ) )
6	3, 4, 5	3bitr4g 303	1 |- ( A. x ( ph <-> ps ) -> ( F/ x ph <-> F/ x ps ) )

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

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

This theorem is referenced by:  nfbii  1778