Theorem hb3an 1830

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.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Hypotheses

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

Assertion

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

Proof of Theorem hb3an

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 |- (ph -> A.xph)
2	1	nfi 1551	. . 3 |- F/xph
3		hb.2	. . . 4 |- (ps -> A.xps)
4	3	nfi 1551	. . 3 |- F/xps
5		hb.3	. . . 4 |- (ch -> A.xch)
6	5	nfi 1551	. . 3 |- F/xch
7	2, 4, 6	nf3an 1827	. 2 |- F/x(ph /\ ps /\ ch)
8	7	nfri 1762	1 |- ((ph /\ ps /\ ch) -> A.x(ph /\ ps /\ ch))

Syntax hints:   -> wi 4   /\ w3a 934  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)