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Theorem df3nandALT2 31051
Description: The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
Assertion
Ref Expression
df3nandALT2  |-  ( (
ph  -/\  ps  -/\  ch )  <->  -.  ( ph  /\  ps  /\ 
ch ) )

Proof of Theorem df3nandALT2
StepHypRef Expression
1 df-3nand 31049 . 2  |-  ( (
ph  -/\  ps  -/\  ch )  <->  (
ph  ->  ( ps  ->  -. 
ch ) ) )
2 imnan 423 . . 3  |-  ( ( ps  ->  -.  ch )  <->  -.  ( ps  /\  ch ) )
32imbi2i 313 . 2  |-  ( (
ph  ->  ( ps  ->  -. 
ch ) )  <->  ( ph  ->  -.  ( ps  /\  ch ) ) )
4 imnan 423 . . 3  |-  ( (
ph  ->  -.  ( ps  /\ 
ch ) )  <->  -.  ( ph  /\  ( ps  /\  ch ) ) )
5 3anass 986 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ph  /\  ( ps  /\  ch ) ) )
64, 5xchbinxr 312 . 2  |-  ( (
ph  ->  -.  ( ps  /\ 
ch ) )  <->  -.  ( ph  /\  ps  /\  ch ) )
71, 3, 63bitri 274 1  |-  ( (
ph  -/\  ps  -/\  ch )  <->  -.  ( ph  /\  ps  /\ 
ch ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    -/\ w3nand 31048
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-3an 984  df-3nand 31049
This theorem is referenced by: (None)
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