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Theorem mpto2 1524
Description: Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or  \/_. See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.)
Hypotheses
Ref Expression
mpto2.1  |-  ph
mpto2.2  |-  ( ph  \/_ 
ps )
Assertion
Ref Expression
mpto2  |-  -.  ps

Proof of Theorem mpto2
StepHypRef Expression
1 mpto2.1 . 2  |-  ph
2 mpto2.2 . . . 4  |-  ( ph  \/_ 
ps )
3 df-xor 1296 . . . 4  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
42, 3mpbi 199 . . 3  |-  -.  ( ph 
<->  ps )
5 xor3 346 . . 3  |-  ( -.  ( ph  <->  ps )  <->  (
ph 
<->  -.  ps ) )
64, 5mpbi 199 . 2  |-  ( ph  <->  -. 
ps )
71, 6mpbi 199 1  |-  -.  ps
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/_ wxo 1295
This theorem is referenced by:  mtp-xor  1526
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 177  df-xor 1296
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