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Theorem nfand 1501
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 1391 . . . . . 6  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
5 df-nf 1391 . . . . . 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 1411 . . . . 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 1411 . . . . . 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 1385 . . . 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 1391 . 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 1283   F/wnf 1390
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-nf 1391
This theorem is referenced by:  nf3and  1502  nfbid  1521  nfsbxy  1861  nfsbxyt  1862  nfeld  2238  nfrexdxy  2405  nfreudxy  2533  nfifd  3398  nfriotadxy  5555  bdsepnft  11121
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