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Theorem bijust 175
Description: Theorem used to justify definition of biconditional df-bi 177. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
Assertion
Ref Expression
bijust  |-  -.  (
( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )  ->  -.  ( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) ) )

Proof of Theorem bijust
StepHypRef Expression
1 id 19 . 2  |-  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
2 pm2.01 160 . 2  |-  ( ( ( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )  ->  -.  ( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) ) )  ->  -.  ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) ) )
31, 2mt2 170 1  |-  -.  (
( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )  ->  -.  ( -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
)  ->  -.  (
( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
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