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

Theorem bija 355
Description: Combine antecedents into a single biconditional. This inference, reminiscent of ja 161, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 238 and pm5.21im 349). (Contributed by Wolf Lammen, 13-May-2013.)
Hypotheses
Ref Expression
bija.1  |-  ( ph  ->  ( ps  ->  ch ) )
bija.2  |-  ( -. 
ph  ->  ( -.  ps  ->  ch ) )
Assertion
Ref Expression
bija  |-  ( (
ph 
<->  ps )  ->  ch )

Proof of Theorem bija
StepHypRef Expression
1 bi2 198 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
2 bija.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
31, 2syli 37 . 2  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ch ) )
4 bi1 186 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
54con3d 133 . . 3  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  -.  ph )
)
6 bija.2 . . 3  |-  ( -. 
ph  ->  ( -.  ps  ->  ch ) )
75, 6syli 37 . 2  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  ch )
)
83, 7pm2.61d 158 1  |-  ( (
ph 
<->  ps )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184
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
This theorem is referenced by:  equveli  2061  wl-aleq  29915  wl-nfeqfb  29917  bj-bibibi  33657
  Copyright terms: Public domain W3C validator