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Theorem imnand2 24838
Description: An  -> nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
Assertion
Ref Expression
imnand2  |-  ( ( -.  ph  ->  ps )  <->  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )

Proof of Theorem imnand2
StepHypRef Expression
1 nannot 1293 . . . 4  |-  ( -. 
ph 
<->  ( ph  -/\  ph )
)
2 nannot 1293 . . . 4  |-  ( -. 
ps 
<->  ( ps  -/\  ps )
)
31, 2anbi12i 678 . . 3  |-  ( ( -.  ph  /\  -.  ps ) 
<->  ( ( ph  -/\  ph )  /\  ( ps  -/\  ps )
) )
43notbii 287 . 2  |-  ( -.  ( -.  ph  /\  -.  ps )  <->  -.  (
( ph  -/\  ph )  /\  ( ps  -/\  ps )
) )
5 iman 413 . 2  |-  ( ( -.  ph  ->  ps )  <->  -.  ( -.  ph  /\  -.  ps ) )
6 df-nan 1288 . 2  |-  ( ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
)  <->  -.  ( ( ph  -/\  ph )  /\  ( ps  -/\  ps ) ) )
74, 5, 63bitr4i 268 1  |-  ( ( -.  ph  ->  ps )  <->  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    -/\ wnan 1287
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
  Copyright terms: Public domain W3C validator