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Theorem andnand1 24909
Description: Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
Assertion
Ref Expression
andnand1  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  -/\  ps  -/\  ch )  -/\  ( ph  -/\ 
ps  -/\  ch ) ) )

Proof of Theorem andnand1
StepHypRef Expression
1 3anass 938 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ph  /\  ( ps  /\  ch ) ) )
2 pm4.63 410 . . . 4  |-  ( -.  ( ps  ->  -.  ch )  <->  ( ps  /\  ch ) )
32anbi2i 675 . . 3  |-  ( (
ph  /\  -.  ( ps  ->  -.  ch )
)  <->  ( ph  /\  ( ps  /\  ch )
) )
4 annim 414 . . 3  |-  ( (
ph  /\  -.  ( ps  ->  -.  ch )
)  <->  -.  ( ph  ->  ( ps  ->  -.  ch ) ) )
51, 3, 43bitr2i 264 . 2  |-  ( (
ph  /\  ps  /\  ch ) 
<->  -.  ( ph  ->  ( ps  ->  -.  ch )
) )
6 df-3nand 24906 . . 3  |-  ( (
ph  -/\  ps  -/\  ch )  <->  (
ph  ->  ( ps  ->  -. 
ch ) ) )
76notbii 287 . 2  |-  ( -.  ( ph  -/\  ps  -/\  ch )  <->  -.  ( ph  ->  ( ps  ->  -.  ch ) ) )
8 nannot 1293 . 2  |-  ( -.  ( ph  -/\  ps  -/\  ch )  <->  ( ( ph  -/\ 
ps  -/\  ch )  -/\  ( ph  -/\  ps  -/\  ch )
) )
95, 7, 83bitr2i 264 1  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  -/\  ps  -/\  ch )  -/\  ( ph  -/\ 
ps  -/\  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    -/\ wnan 1287    -/\ w3nand 24905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-nan 1288  df-3nand 24906
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