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Theorem nannan 1385
Description: Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
Assertion
Ref Expression
nannan  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  ( ph  ->  ( ch  /\  ps ) ) )

Proof of Theorem nannan
StepHypRef Expression
1 imnan 424 . 2  |-  ( (
ph  ->  -.  ( ch  -/\ 
ps ) )  <->  -.  ( ph  /\  ( ch  -/\  ps ) ) )
2 nanan 1382 . . 3  |-  ( ( ch  /\  ps )  <->  -.  ( ch  -/\  ps )
)
32imbi2i 314 . 2  |-  ( (
ph  ->  ( ch  /\  ps ) )  <->  ( ph  ->  -.  ( ch  -/\  ps ) ) )
4 df-nan 1381 . 2  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  -.  ( ph  /\  ( ch  -/\  ps ) ) )
51, 3, 43bitr4ri 282 1  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  <->  ( ph  ->  ( ch  /\  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    -/\ wnan 1380
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 189  df-an 373  df-nan 1381
This theorem is referenced by:  nanim  1387  nanbi  1389  nanbiOLD  1390  nic-mp  1551  nic-ax  1553  waj-ax  31073  lukshef-ax2  31074  arg-ax  31075  rp-fakenanass  36085
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