MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hb3an Structured version   Visualization version   Unicode version

Theorem hb3an 2035
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. 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
StepHypRef Expression
1 hb.1 . . . 4  |-  ( ph  ->  A. x ph )
21nfi 1682 . . 3  |-  F/ x ph
3 hb.2 . . . 4  |-  ( ps 
->  A. x ps )
43nfi 1682 . . 3  |-  F/ x ps
5 hb.3 . . . 4  |-  ( ch 
->  A. x ch )
65nfi 1682 . . 3  |-  F/ x ch
72, 4, 6nf3an 2033 . 2  |-  F/ x
( ph  /\  ps  /\  ch )
87nfri 1972 1  |-  ( (
ph  /\  ps  /\  ch )  ->  A. x ( ph  /\ 
ps  /\  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007   A.wal 1450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-ex 1672  df-nf 1676
This theorem is referenced by:  bnj982  29662  bnj1276  29698  bnj1350  29709
  Copyright terms: Public domain W3C validator