ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfand Unicode version

Theorem nfand 1505
Description: If in a context  x is not free in  ps and  ch, it is not free in  ( ps  /\  ch ). (Contributed by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfand.1  |-  ( ph  ->  F/ x ps )
nfand.2  |-  ( ph  ->  F/ x ch )
Assertion
Ref Expression
nfand  |-  ( ph  ->  F/ x ( ps 
/\  ch ) )

Proof of Theorem nfand
StepHypRef Expression
1 nfand.1 . . . 4  |-  ( ph  ->  F/ x ps )
2 nfand.2 . . . 4  |-  ( ph  ->  F/ x ch )
31, 2jca 300 . . 3  |-  ( ph  ->  ( F/ x ps 
/\  F/ x ch ) )
4 df-nf 1395 . . . . . 6  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
5 df-nf 1395 . . . . . 6  |-  ( F/ x ch  <->  A. x
( ch  ->  A. x ch ) )
64, 5anbi12i 448 . . . . 5  |-  ( ( F/ x ps  /\  F/ x ch )  <->  ( A. x ( ps  ->  A. x ps )  /\  A. x ( ch  ->  A. x ch ) ) )
7 19.26 1415 . . . . 5  |-  ( A. x ( ( ps 
->  A. x ps )  /\  ( ch  ->  A. x ch ) )  <->  ( A. x ( ps  ->  A. x ps )  /\  A. x ( ch  ->  A. x ch ) ) )
86, 7bitr4i 185 . . . 4  |-  ( ( F/ x ps  /\  F/ x ch )  <->  A. x
( ( ps  ->  A. x ps )  /\  ( ch  ->  A. x ch ) ) )
9 prth 336 . . . . . 6  |-  ( ( ( ps  ->  A. x ps )  /\  ( ch  ->  A. x ch )
)  ->  ( ( ps  /\  ch )  -> 
( A. x ps 
/\  A. x ch )
) )
10 19.26 1415 . . . . . 6  |-  ( A. x ( ps  /\  ch )  <->  ( A. x ps  /\  A. x ch ) )
119, 10syl6ibr 160 . . . . 5  |-  ( ( ( ps  ->  A. x ps )  /\  ( ch  ->  A. x ch )
)  ->  ( ( ps  /\  ch )  ->  A. x ( ps  /\  ch ) ) )
1211alimi 1389 . . . 4  |-  ( A. x ( ( ps 
->  A. x ps )  /\  ( ch  ->  A. x ch ) )  ->  A. x
( ( ps  /\  ch )  ->  A. x
( ps  /\  ch ) ) )
138, 12sylbi 119 . . 3  |-  ( ( F/ x ps  /\  F/ x ch )  ->  A. x ( ( ps 
/\  ch )  ->  A. x
( ps  /\  ch ) ) )
143, 13syl 14 . 2  |-  ( ph  ->  A. x ( ( ps  /\  ch )  ->  A. x ( ps 
/\  ch ) ) )
15 df-nf 1395 . 2  |-  ( F/ x ( ps  /\  ch )  <->  A. x ( ( ps  /\  ch )  ->  A. x ( ps 
/\  ch ) ) )
1614, 15sylibr 132 1  |-  ( ph  ->  F/ x ( ps 
/\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287   F/wnf 1394
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 1381  ax-gen 1383
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  nf3and  1506  nfbid  1525  nfsbxy  1866  nfsbxyt  1867  nfeld  2244  nfrexdxy  2411  nfreudxy  2540  nfifd  3414  nfriotadxy  5598  bdsepnft  11435
  Copyright terms: Public domain W3C validator