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Theorem nic-luk3 1572
Description: Proof of luk-3 1536 from nic-ax 1552 and nic-mp 1550. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nic-luk3  |-  ( ph  ->  ( -.  ph  ->  ps ) )

Proof of Theorem nic-luk3
StepHypRef Expression
1 nic-dfim 1548 . . . 4  |-  ( ( ( -.  ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( -.  ph  ->  ps ) )  -/\  ( ( ( -. 
ph  -/\  ( ps  -/\  ps ) )  -/\  ( -.  ph  -/\  ( ps  -/\ 
ps ) ) ) 
-/\  ( ( -. 
ph  ->  ps )  -/\  ( -.  ph  ->  ps ) ) ) )
21nic-bi1 1567 . . 3  |-  ( ( -.  ph  -/\  ( ps 
-/\  ps ) )  -/\  ( ( -.  ph  ->  ps )  -/\  ( -.  ph  ->  ps )
) )
3 nic-dfneg 1549 . . . . 5  |-  ( ( ( ph  -/\  ph )  -/\  -.  ph )  -/\  ( ( ( ph  -/\  ph )  -/\  ( ph  -/\  ph ) )  -/\  ( -.  ph  -/\  -.  ph )
) )
43nic-bi2 1568 . . . 4  |-  ( -. 
ph  -/\  ( ( ph  -/\  ph )  -/\  ( ph  -/\  ph ) ) )
5 nic-id 1557 . . . 4  |-  ( ph  -/\  ( ph  -/\  ph )
)
64, 5nic-iimp1 1561 . . 3  |-  ( ph  -/\ 
-.  ph )
72, 6nic-iimp2 1562 . 2  |-  ( ph  -/\  ( ( -.  ph  ->  ps )  -/\  ( -.  ph  ->  ps )
) )
8 nic-dfim 1548 . . 3  |-  ( ( ( ph  -/\  (
( -.  ph  ->  ps )  -/\  ( -.  ph 
->  ps ) ) ) 
-/\  ( ph  ->  ( -.  ph  ->  ps )
) )  -/\  (
( ( ph  -/\  (
( -.  ph  ->  ps )  -/\  ( -.  ph 
->  ps ) ) ) 
-/\  ( ph  -/\  (
( -.  ph  ->  ps )  -/\  ( -.  ph 
->  ps ) ) ) )  -/\  ( ( ph  ->  ( -.  ph  ->  ps ) )  -/\  ( ph  ->  ( -.  ph 
->  ps ) ) ) ) )
98nic-bi1 1567 . 2  |-  ( (
ph  -/\  ( ( -. 
ph  ->  ps )  -/\  ( -.  ph  ->  ps ) ) )  -/\  ( ( ph  ->  ( -.  ph  ->  ps )
)  -/\  ( ph  ->  ( -.  ph  ->  ps ) ) ) )
107, 9nic-mp 1550 1  |-  ( ph  ->  ( -.  ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    -/\ wnan 1379
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 188  df-or 371  df-an 372  df-nan 1380
This theorem is referenced by: (None)
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