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Theorem tsbi3 29070
Description: A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsbi3  |-  ( th 
->  ( ( ph  \/  -.  ps )  \/  -.  ( ph  <->  ps ) ) )

Proof of Theorem tsbi3
StepHypRef Expression
1 orc 385 . . 3  |-  ( (
ph  \/  -.  ps )  ->  ( ( ph  \/  -.  ps )  \/  -.  ( ph  <->  ps ) ) )
2 bi2 198 . . . . . 6  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
3 con34b 292 . . . . . . 7  |-  ( ( ps  ->  ph )  <->  ( -.  ph 
->  -.  ps ) )
4 df-or 370 . . . . . . . 8  |-  ( (
ph  \/  -.  ps )  <->  ( -.  ph  ->  -.  ps ) )
54biimpri 206 . . . . . . 7  |-  ( ( -.  ph  ->  -.  ps )  ->  ( ph  \/  -.  ps ) )
63, 5sylbi 195 . . . . . 6  |-  ( ( ps  ->  ph )  -> 
( ph  \/  -.  ps ) )
72, 6syl 16 . . . . 5  |-  ( (
ph 
<->  ps )  ->  ( ph  \/  -.  ps )
)
87con3i 135 . . . 4  |-  ( -.  ( ph  \/  -.  ps )  ->  -.  ( ph 
<->  ps ) )
98olcd 393 . . 3  |-  ( -.  ( ph  \/  -.  ps )  ->  ( (
ph  \/  -.  ps )  \/  -.  ( ph  <->  ps )
) )
101, 9pm2.61i 164 . 2  |-  ( (
ph  \/  -.  ps )  \/  -.  ( ph  <->  ps )
)
1110a1i 11 1  |-  ( th 
->  ( ( ph  \/  -.  ps )  \/  -.  ( ph  <->  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368
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-or 370
This theorem is referenced by:  tsbi4  29071  tsxo3  29074  mpt2bi123f  29099  mptbi12f  29103  ac6s6  29108
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