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Theorem merco2 1496
Description: A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1473. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
merco2  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( et  ->  ph ) ) ) )

Proof of Theorem merco2
StepHypRef Expression
1 falim 1325 . . . . . 6  |-  (  F. 
->  ch )
2 pm2.04 78 . . . . . 6  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( (  F.  ->  ch )  -> 
( ( ph  ->  ps )  ->  th )
) )
31, 2mpi 18 . . . . 5  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( ( ph  ->  ps )  ->  th ) )
4 jarl 157 . . . . . 6  |-  ( ( ( ph  ->  ps )  ->  th )  ->  ( -.  ph  ->  th )
)
5 idd 23 . . . . . 6  |-  ( ( ( ph  ->  ps )  ->  th )  ->  ( th  ->  th ) )
64, 5jad 156 . . . . 5  |-  ( ( ( ph  ->  ps )  ->  th )  ->  (
( ph  ->  th )  ->  th ) )
7 looinv 176 . . . . 5  |-  ( ( ( ph  ->  th )  ->  th )  ->  (
( th  ->  ph )  ->  ph ) )
83, 6, 73syl 20 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( ( th  ->  ph )  ->  ph )
)
98a1dd 44 . . 3  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( ( th  ->  ph )  ->  ( ta  ->  ph ) ) )
109a1i 12 . 2  |-  ( et 
->  ( ( ( ph  ->  ps )  ->  (
(  F.  ->  ch )  ->  th ) )  -> 
( ( th  ->  ph )  ->  ( ta  ->  ph ) ) ) )
1110com4l 80 1  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( ( th  ->  ph )  ->  ( ta  ->  ( et  ->  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    F. wfal 1313
This theorem is referenced by:  mercolem1  1497  mercolem2  1498  mercolem3  1499  mercolem4  1500  mercolem5  1501  mercolem6  1502  mercolem7  1503  mercolem8  1504  re1tbw4  1508
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-tru 1315  df-fal 1316
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