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Theorem df3nandALT2 25861
Description: The double nand expressed in terms of negation and and. (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 25859 . 2  |-  ( (
ph  -/\  ps  -/\  ch )  <->  (
ph  ->  ( ps  ->  -. 
ch ) ) )
2 imnan 412 . . 3  |-  ( ( ps  ->  -.  ch )  <->  -.  ( ps  /\  ch ) )
32imbi2i 304 . 2  |-  ( (
ph  ->  ( ps  ->  -. 
ch ) )  <->  ( ph  ->  -.  ( ps  /\  ch ) ) )
4 imnan 412 . . 3  |-  ( (
ph  ->  -.  ( ps  /\ 
ch ) )  <->  -.  ( ph  /\  ( ps  /\  ch ) ) )
5 3anass 940 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ph  /\  ( ps  /\  ch ) ) )
64, 5xchbinxr 303 . 2  |-  ( (
ph  ->  -.  ( ps  /\ 
ch ) )  <->  -.  ( ph  /\  ps  /\  ch ) )
71, 3, 63bitri 263 1  |-  ( (
ph  -/\  ps  -/\  ch )  <->  -.  ( ph  /\  ps  /\ 
ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    -/\ w3nand 25858
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-3nand 25859
  Copyright terms: Public domain W3C validator