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Theorem imnand2 26154
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 1303 . . . 4  |-  ( -. 
ph 
<->  ( ph  -/\  ph )
)
2 nannot 1303 . . . 4  |-  ( -. 
ps 
<->  ( ps  -/\  ps )
)
31, 2anbi12i 680 . . 3  |-  ( ( -.  ph  /\  -.  ps ) 
<->  ( ( ph  -/\  ph )  /\  ( ps  -/\  ps )
) )
43notbii 289 . 2  |-  ( -.  ( -.  ph  /\  -.  ps )  <->  -.  (
( ph  -/\  ph )  /\  ( ps  -/\  ps )
) )
5 iman 415 . 2  |-  ( ( -.  ph  ->  ps )  <->  -.  ( -.  ph  /\  -.  ps ) )
6 df-nan 1298 . 2  |-  ( ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
)  <->  -.  ( ( ph  -/\  ph )  /\  ( ps  -/\  ps ) ) )
74, 5, 63bitr4i 270 1  |-  ( ( -.  ph  ->  ps )  <->  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    -/\ wnan 1297
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 179  df-an 362  df-nan 1298
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