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Theorem andnand1 26140
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 940 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ph  /\  ( ps  /\  ch ) ) )
2 pm4.63 411 . . . 4  |-  ( -.  ( ps  ->  -.  ch )  <->  ( ps  /\  ch ) )
32anbi2i 676 . . 3  |-  ( (
ph  /\  -.  ( ps  ->  -.  ch )
)  <->  ( ph  /\  ( ps  /\  ch )
) )
4 annim 415 . . 3  |-  ( (
ph  /\  -.  ( ps  ->  -.  ch )
)  <->  -.  ( ph  ->  ( ps  ->  -.  ch ) ) )
51, 3, 43bitr2i 265 . 2  |-  ( (
ph  /\  ps  /\  ch ) 
<->  -.  ( ph  ->  ( ps  ->  -.  ch )
) )
6 df-3nand 26137 . . 3  |-  ( (
ph  -/\  ps  -/\  ch )  <->  (
ph  ->  ( ps  ->  -. 
ch ) ) )
76notbii 288 . 2  |-  ( -.  ( ph  -/\  ps  -/\  ch )  <->  -.  ( ph  ->  ( ps  ->  -.  ch ) ) )
8 nannot 1302 . 2  |-  ( -.  ( ph  -/\  ps  -/\  ch )  <->  ( ( ph  -/\ 
ps  -/\  ch )  -/\  ( ph  -/\  ps  -/\  ch )
) )
95, 7, 83bitr2i 265 1  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  -/\  ps  -/\  ch )  -/\  ( ph  -/\ 
ps  -/\  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    -/\ wnan 1296    -/\ w3nand 26136
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 178  df-an 361  df-3an 938  df-nan 1297  df-3nand 26137
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