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Theorem bija 344
Description: Combine antecedents into a single bi-conditional. This inference, reminiscent of ja 153, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 229 and pm5.21im 338). (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 189 . . 3  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
2 bija.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
31, 2syli 33 . 2  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ch ) )
4 bi1 178 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
54con3d 125 . . 3  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  -.  ph )
)
6 bija.2 . . 3  |-  ( -. 
ph  ->  ( -.  ps  ->  ch ) )
75, 6syli 33 . 2  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  ch )
)
83, 7pm2.61d 150 1  |-  ( (
ph 
<->  ps )  ->  ch )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176
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
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