MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nic-stdmp Unicode version

Theorem nic-stdmp 1445
Description: Derive the standard modus ponens from nic-mp 1426. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nic-smin  |-  ph
nic-smaj  |-  ( ph  ->  ps )
Assertion
Ref Expression
nic-stdmp  |-  ps

Proof of Theorem nic-stdmp
StepHypRef Expression
1 nic-smin . 2  |-  ph
2 nic-smaj . . 3  |-  ( ph  ->  ps )
3 nic-dfim 1424 . . . 4  |-  ( ( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  ->  ps ) )  -/\  (
( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  -/\  ( ps  -/\  ps ) ) )  -/\  ( ( ph  ->  ps )  -/\  ( ph  ->  ps )
) ) )
43nic-bi2 1444 . . 3  |-  ( (
ph  ->  ps )  -/\  ( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  -/\  ( ps  -/\  ps ) ) ) )
52, 4nic-mp 1426 . 2  |-  ( ph  -/\  ( ps  -/\  ps )
)
61, 5nic-mp 1426 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    -/\ 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-or 359  df-an 360  df-nan 1288
  Copyright terms: Public domain W3C validator