Theorem hb3an 1483

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 922	. 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 1480	. . 3 |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )
5		hb.3	. . 3 |- ( ch -> A. x ch )
6	4, 5	hban 1480	. 2 |- ( ( ( ph /\ ps ) /\ ch ) -> A. x ( ( ph /\ ps ) /\ ch ) )
7	1, 6	hbxfrbi 1402	1 |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   A.wal 1283

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379

This theorem depends on definitions:  df-bi 115  df-3an 922

This theorem is referenced by: (None)