Theorem hbbid 1575

Description: Deduction form of bound-variable hypothesis builder hbbi 1548. (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 1573	. . 3 |- ( ph -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) )
5	1, 3, 2	hbimd 1573	. . 3 |- ( ph -> ( ( ch -> ps ) -> A. x ( ch -> ps ) ) )
6	4, 5	anim12d 335	. 2 |- ( ph -> ( ( ( ps -> ch ) /\ ( ch -> ps ) ) -> ( A. x ( ps -> ch ) /\ A. x ( ch -> ps ) ) ) )
7		dfbi2 388	. 2 |- ( ( ps <-> ch ) <-> ( ( ps -> ch ) /\ ( ch -> ps ) ) )
8		albiim 1487	. 2 |- ( A. x ( ps <-> ch ) <-> ( A. x ( ps -> ch ) /\ A. x ( ch -> ps ) ) )
9	6, 7, 8	3imtr4g 205	1 |- ( ph -> ( ( ps <-> ch ) -> A. x ( ps <-> ch ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-i5r 1535

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)