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Theorem merlem10 1406
Description: Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merlem10  |-  ( (
ph  ->  ( ph  ->  ps ) )  ->  ( th  ->  ( ph  ->  ps ) ) )

Proof of Theorem merlem10
StepHypRef Expression
1 ax-meredith 1396 . 2  |-  ( ( ( ( ( ph  ->  ph )  ->  ( -.  ph  ->  -.  ph )
)  ->  ph )  ->  ph )  ->  ( (
ph  ->  ph )  ->  ( ph  ->  ph ) ) )
2 ax-meredith 1396 . . 3  |-  ( ( ( ( ( (
ph  ->  ps )  ->  ph )  ->  ( -. 
ph  ->  -.  th )
)  ->  ph )  ->  ph )  ->  ( (
ph  ->  ( ph  ->  ps ) )  ->  ( th  ->  ( ph  ->  ps ) ) ) )
3 merlem9 1405 . . 3  |-  ( ( ( ( ( ( ( ph  ->  ps )  ->  ph )  ->  ( -.  ph  ->  -.  th )
)  ->  ph )  ->  ph )  ->  ( (
ph  ->  ( ph  ->  ps ) )  ->  ( th  ->  ( ph  ->  ps ) ) ) )  ->  ( ( ( ( ( ( ph  ->  ph )  ->  ( -.  ph  ->  -.  ph )
)  ->  ph )  ->  ph )  ->  ( (
ph  ->  ph )  ->  ( ph  ->  ph ) ) )  ->  ( ( ph  ->  ( ph  ->  ps ) )  ->  ( th  ->  ( ph  ->  ps ) ) ) ) )
42, 3ax-mp 8 . 2  |-  ( ( ( ( ( (
ph  ->  ph )  ->  ( -.  ph  ->  -.  ph )
)  ->  ph )  ->  ph )  ->  ( (
ph  ->  ph )  ->  ( ph  ->  ph ) ) )  ->  ( ( ph  ->  ( ph  ->  ps ) )  ->  ( th  ->  ( ph  ->  ps ) ) ) )
51, 4ax-mp 8 1  |-  ( (
ph  ->  ( ph  ->  ps ) )  ->  ( th  ->  ( ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  merlem11  1407
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
  Copyright terms: Public domain W3C validator