MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mp2ani Structured version   Unicode version

Theorem mp2ani 673
Description: An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
Hypotheses
Ref Expression
mp2ani.1  |-  ps
mp2ani.2  |-  ch
mp2ani.3  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
Assertion
Ref Expression
mp2ani  |-  ( ph  ->  th )

Proof of Theorem mp2ani
StepHypRef Expression
1 mp2ani.2 . 2  |-  ch
2 mp2ani.1 . . 3  |-  ps
3 mp2ani.3 . . 3  |-  ( ph  ->  ( ( ps  /\  ch )  ->  th )
)
42, 3mpani 671 . 2  |-  ( ph  ->  ( ch  ->  th )
)
51, 4mpi 17 1  |-  ( ph  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369
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 185  df-an 371
This theorem is referenced by:  th3q  7205  dfom3  7849  dfac5lem4  8292  dfac9  8301  cflem  8411  canthp1lem2  8816  gcdaddmlem  13708  sto1i  25575  stji1i  25581  cnvoprab  25958  kur14lem9  27032  dfon2lem4  27528
  Copyright terms: Public domain W3C validator