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Theorem merco2 1510
Description: A single axiom for propositional calculus offered by Meredith.

This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1487. (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 1337 . . . . . 6  |-  (  F. 
->  ch )
2 pm2.04 78 . . . . . 6  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( (  F.  ->  ch )  -> 
( ( ph  ->  ps )  ->  th )
) )
31, 2mpi 17 . . . . 5  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( ( ph  ->  ps )  ->  th ) )
4 jarl 157 . . . . . 6  |-  ( ( ( ph  ->  ps )  ->  th )  ->  ( -.  ph  ->  th )
)
5 idd 22 . . . . . 6  |-  ( ( ( ph  ->  ps )  ->  th )  ->  ( th  ->  th ) )
64, 5jad 156 . . . . 5  |-  ( ( ( ph  ->  ps )  ->  th )  ->  (
( ph  ->  th )  ->  th ) )
7 looinv 175 . . . . 5  |-  ( ( ( ph  ->  th )  ->  th )  ->  (
( th  ->  ph )  ->  ph ) )
83, 6, 73syl 19 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( ( th  ->  ph )  ->  ph )
)
98a1dd 44 . . 3  |-  ( ( ( ph  ->  ps )  ->  ( (  F. 
->  ch )  ->  th )
)  ->  ( ( th  ->  ph )  ->  ( ta  ->  ph ) ) )
109a1i 11 . 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 4    F. wfal 1326
This theorem is referenced by:  mercolem1  1511  mercolem2  1512  mercolem3  1513  mercolem4  1514  mercolem5  1515  mercolem6  1516  mercolem7  1517  mercolem8  1518  re1tbw4  1522
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 178  df-tru 1328  df-fal 1329
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