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Theorem nannan 1291
Description: Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.)
Assertion
Ref Expression
nannan  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  ( ph  ->  ( ch  /\  ps ) ) )

Proof of Theorem nannan
StepHypRef Expression
1 df-nan 1288 . . 3  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  -.  ( ph  /\  ( ch  -/\  ps ) ) )
2 df-nan 1288 . . . 4  |-  ( ( ch  -/\  ps )  <->  -.  ( ch  /\  ps ) )
32anbi2i 675 . . 3  |-  ( (
ph  /\  ( ch  -/\ 
ps ) )  <->  ( ph  /\ 
-.  ( ch  /\  ps ) ) )
41, 3xchbinx 301 . 2  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  -.  ( ph  /\  -.  ( ch 
/\  ps ) ) )
5 iman 413 . 2  |-  ( (
ph  ->  ( ch  /\  ps ) )  <->  -.  ( ph  /\  -.  ( ch 
/\  ps ) ) )
64, 5bitr4i 243 1  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  ( ph  ->  ( ch  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    -/\ wnan 1287
This theorem is referenced by:  nanim  1292  nic-mp  1426  nic-ax  1428  waj-ax  24264  lukshef-ax2  24265  arg-ax  24266
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-nan 1288
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