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Theorem df3nandALT1 31062
Description: The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
Assertion
Ref Expression
df3nandALT1  |-  ( (
ph  -/\  ps  -/\  ch )  <->  (
ph  -/\  ( ( ps 
-/\  ch )  -/\  ( ps  -/\  ch ) ) ) )

Proof of Theorem df3nandALT1
StepHypRef Expression
1 iman 425 . . 3  |-  ( (
ph  ->  ( ( ps 
-/\  ch )  /\  ( ps  -/\  ch ) ) )  <->  -.  ( ph  /\ 
-.  ( ( ps 
-/\  ch )  /\  ( ps  -/\  ch ) ) ) )
2 imnan 423 . . . . . . . 8  |-  ( ( ps  ->  -.  ch )  <->  -.  ( ps  /\  ch ) )
32biimpi 197 . . . . . . 7  |-  ( ( ps  ->  -.  ch )  ->  -.  ( ps  /\  ch ) )
43, 3jca 534 . . . . . 6  |-  ( ( ps  ->  -.  ch )  ->  ( -.  ( ps 
/\  ch )  /\  -.  ( ps  /\  ch )
) )
52biimpri 209 . . . . . . 7  |-  ( -.  ( ps  /\  ch )  ->  ( ps  ->  -. 
ch ) )
65adantl 467 . . . . . 6  |-  ( ( -.  ( ps  /\  ch )  /\  -.  ( ps  /\  ch ) )  ->  ( ps  ->  -. 
ch ) )
74, 6impbii 190 . . . . 5  |-  ( ( ps  ->  -.  ch )  <->  ( -.  ( ps  /\  ch )  /\  -.  ( ps  /\  ch ) ) )
8 df-nan 1380 . . . . . 6  |-  ( ( ps  -/\  ch )  <->  -.  ( ps  /\  ch ) )
98, 8anbi12i 701 . . . . 5  |-  ( ( ( ps  -/\  ch )  /\  ( ps  -/\  ch )
)  <->  ( -.  ( ps  /\  ch )  /\  -.  ( ps  /\  ch ) ) )
107, 9bitr4i 255 . . . 4  |-  ( ( ps  ->  -.  ch )  <->  ( ( ps  -/\  ch )  /\  ( ps  -/\  ch )
) )
1110imbi2i 313 . . 3  |-  ( (
ph  ->  ( ps  ->  -. 
ch ) )  <->  ( ph  ->  ( ( ps  -/\  ch )  /\  ( ps 
-/\  ch ) ) ) )
12 df-nan 1380 . . . . 5  |-  ( ( ( ps  -/\  ch )  -/\  ( ps  -/\  ch )
)  <->  -.  ( ( ps  -/\  ch )  /\  ( ps  -/\  ch )
) )
1312anbi2i 698 . . . 4  |-  ( (
ph  /\  ( ( ps  -/\  ch )  -/\  ( ps  -/\  ch )
) )  <->  ( ph  /\ 
-.  ( ( ps 
-/\  ch )  /\  ( ps  -/\  ch ) ) ) )
1413notbii 297 . . 3  |-  ( -.  ( ph  /\  (
( ps  -/\  ch )  -/\  ( ps  -/\  ch )
) )  <->  -.  ( ph  /\  -.  ( ( ps  -/\  ch )  /\  ( ps  -/\  ch )
) ) )
151, 11, 143bitr4i 280 . 2  |-  ( (
ph  ->  ( ps  ->  -. 
ch ) )  <->  -.  ( ph  /\  ( ( ps 
-/\  ch )  -/\  ( ps  -/\  ch ) ) ) )
16 df-3nand 31061 . 2  |-  ( (
ph  -/\  ps  -/\  ch )  <->  (
ph  ->  ( ps  ->  -. 
ch ) ) )
17 df-nan 1380 . 2  |-  ( (
ph  -/\  ( ( ps 
-/\  ch )  -/\  ( ps  -/\  ch ) ) )  <->  -.  ( ph  /\  ( ( ps  -/\  ch )  -/\  ( ps  -/\ 
ch ) ) ) )
1815, 16, 173bitr4i 280 1  |-  ( (
ph  -/\  ps  -/\  ch )  <->  (
ph  -/\  ( ( ps 
-/\  ch )  -/\  ( ps  -/\  ch ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    -/\ wnan 1379    -/\ w3nand 31060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-an 372  df-nan 1380  df-3nand 31061
This theorem is referenced by: (None)
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