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Theorem tsim1 32279
Description: A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsim1  |-  ( th 
->  ( ( -.  ph  \/  ps )  \/  -.  ( ph  ->  ps )
) )

Proof of Theorem tsim1
StepHypRef Expression
1 exmid 416 . . 3  |-  ( (
ph  ->  ps )  \/ 
-.  ( ph  ->  ps ) )
2 df-or 371 . . . . 5  |-  ( ( -.  ph  \/  ps ) 
<->  ( -.  -.  ph  ->  ps ) )
3 notnot 292 . . . . . . 7  |-  ( ph  <->  -. 
-.  ph )
43bicomi 205 . . . . . 6  |-  ( -. 
-.  ph  <->  ph )
54imbi1i 326 . . . . 5  |-  ( ( -.  -.  ph  ->  ps )  <->  ( ph  ->  ps ) )
62, 5bitri 252 . . . 4  |-  ( ( -.  ph  \/  ps ) 
<->  ( ph  ->  ps ) )
76orbi1i 522 . . 3  |-  ( ( ( -.  ph  \/  ps )  \/  -.  ( ph  ->  ps )
)  <->  ( ( ph  ->  ps )  \/  -.  ( ph  ->  ps )
) )
81, 7mpbir 212 . 2  |-  ( ( -.  ph  \/  ps )  \/  -.  ( ph  ->  ps ) )
98a1i 11 1  |-  ( th 
->  ( ( -.  ph  \/  ps )  \/  -.  ( ph  ->  ps )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 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 188  df-or 371
This theorem is referenced by:  mpt2bi123f  32313  ac6s6  32322
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