Theorem hb3and 1490

Description: Deduction form of bound-variable hypothesis builder hb3an 1550. (Contributed by NM, 17-Feb-2013.)

Hypotheses

Ref	Expression
hb3and.1	|- ( ph -> ( ps -> A. x ps ) )
hb3and.2	|- ( ph -> ( ch -> A. x ch ) )
hb3and.3	|- ( ph -> ( th -> A. x th ) )

Assertion

Ref	Expression
hb3and	|- ( ph -> ( ( ps /\ ch /\ th ) -> A. x ( ps /\ ch /\ th ) ) )

Proof of Theorem hb3and

Step	Hyp	Ref	Expression
1		hb3and.1	. . 3 |- ( ph -> ( ps -> A. x ps ) )
2		hb3and.2	. . 3 |- ( ph -> ( ch -> A. x ch ) )
3		hb3and.3	. . 3 |- ( ph -> ( th -> A. x th ) )
4	1, 2, 3	3anim123d 1319	. 2 |- ( ph -> ( ( ps /\ ch /\ th ) -> ( A. x ps /\ A. x ch /\ A. x th ) ) )
5		19.26-3an 1483	. 2 |- ( A. x ( ps /\ ch /\ th ) <-> ( A. x ps /\ A. x ch /\ A. x th ) )
6	4, 5	imbitrrdi 162	1 |- ( ph -> ( ( ps /\ ch /\ th ) -> A. x ( ps /\ ch /\ th ) ) )

Syntax hints:   -> wi 4   /\ w3a 978   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  df-3an 980

This theorem is referenced by: (None)