Theorem hb3an 1538

Description: If x is not free in ph, ps, and ch, it is not free in ( ph /\ ps /\ ch ). (Contributed by NM, 14-Sep-2003.)

Hypotheses

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

Assertion

Ref	Expression
hb3an	|- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )

Proof of Theorem hb3an

Step	Hyp	Ref	Expression
1		df-3an 970	. 2 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
2		hb.1	. . . 4 |- ( ph -> A. x ph )
3		hb.2	. . . 4 |- ( ps -> A. x ps )
4	2, 3	hban 1535	. . 3 |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )
5		hb.3	. . 3 |- ( ch -> A. x ch )
6	4, 5	hban 1535	. 2 |- ( ( ( ph /\ ps ) /\ ch ) -> A. x ( ( ph /\ ps ) /\ ch ) )
7	1, 6	hbxfrbi 1460	1 |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   A.wal 1341

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 1435  ax-gen 1437

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by: (None)