Theorem hbbid 1554

Description: Deduction form of bound-variable hypothesis builder hbbi 1527. (Contributed by NM, 1-Jan-2002.)

Hypotheses

Ref	Expression
hbbid.1	|- ( ph -> A. x ph )
hbbid.2	|- ( ph -> ( ps -> A. x ps ) )
hbbid.3	|- ( ph -> ( ch -> A. x ch ) )

Assertion

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

Proof of Theorem hbbid

Step	Hyp	Ref	Expression
1		hbbid.1	. . . 4 |- ( ph -> A. x ph )
2		hbbid.2	. . . 4 |- ( ph -> ( ps -> A. x ps ) )
3		hbbid.3	. . . 4 |- ( ph -> ( ch -> A. x ch ) )
4	1, 2, 3	hbimd 1552	. . 3 |- ( ph -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) )
5	1, 3, 2	hbimd 1552	. . 3 |- ( ph -> ( ( ch -> ps ) -> A. x ( ch -> ps ) ) )
6	4, 5	anim12d 333	. 2 |- ( ph -> ( ( ( ps -> ch ) /\ ( ch -> ps ) ) -> ( A. x ( ps -> ch ) /\ A. x ( ch -> ps ) ) ) )
7		dfbi2 385	. 2 |- ( ( ps <-> ch ) <-> ( ( ps -> ch ) /\ ( ch -> ps ) ) )
8		albiim 1463	. 2 |- ( A. x ( ps <-> ch ) <-> ( A. x ( ps -> ch ) /\ A. x ( ch -> ps ) ) )
9	6, 7, 8	3imtr4g 204	1 |- ( ph -> ( ( ps <-> ch ) -> A. x ( ps <-> ch ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329

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 1423  ax-gen 1425  ax-4 1487  ax-i5r 1515

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)