Theorem hbbi 1541

Description: If x is not free in ph and ps, it is not free in ( ph <-> ps ). (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
hb.1	|- ( ph -> A. x ph )
hb.2	|- ( ps -> A. x ps )

Assertion

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

Proof of Theorem hbbi

Step	Hyp	Ref	Expression
1		dfbi2 386	. 2 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2		hb.1	. . . 4 |- ( ph -> A. x ph )
3		hb.2	. . . 4 |- ( ps -> A. x ps )
4	2, 3	hbim 1538	. . 3 |- ( ( ph -> ps ) -> A. x ( ph -> ps ) )
5	3, 2	hbim 1538	. . 3 |- ( ( ps -> ph ) -> A. x ( ps -> ph ) )
6	4, 5	hban 1540	. 2 |- ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) )
7	1, 6	hbxfrbi 1465	1 |- ( ( ph <-> ps ) -> A. x ( ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-i5r 1528

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  euf  2024  sb8euh  2042