Theorem hband 1489

Description: Deduction form of bound-variable hypothesis builder hban 1547. (Contributed by NM, 2-Jan-2002.)

Hypotheses

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

Assertion

Ref	Expression
hband	|- ( ph -> ( ( ps /\ ch ) -> A. x ( ps /\ ch ) ) )

Proof of Theorem hband

Step	Hyp	Ref	Expression
1		hband.1	. . 3 |- ( ph -> ( ps -> A. x ps ) )
2		hband.2	. . 3 |- ( ph -> ( ch -> A. x ch ) )
3	1, 2	anim12d 335	. 2 |- ( ph -> ( ( ps /\ ch ) -> ( A. x ps /\ A. x ch ) ) )
4		19.26 1481	. 2 |- ( A. x ( ps /\ ch ) <-> ( A. x ps /\ A. x ch ) )
5	3, 4	syl6ibr 162	1 |- ( ph -> ( ( ps /\ ch ) -> A. x ( ps /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 104   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

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)