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Theorem mpto2 1534
 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 φ
mpto2.2 (φψ)
Assertion
Ref Expression
mpto2 ¬ ψ

Proof of Theorem mpto2
StepHypRef Expression
1 mpto2.1 . 2 φ
2 mpto2.2 . . . 4 (φψ)
3 df-xor 1305 . . . 4 ((φψ) ↔ ¬ (φψ))
42, 3mpbi 199 . . 3 ¬ (φψ)
5 xor3 346 . . 3 (¬ (φψ) ↔ (φ ↔ ¬ ψ))
64, 5mpbi 199 . 2 (φ ↔ ¬ ψ)
71, 6mpbi 199 1 ¬ ψ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ⊻ wxo 1304 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 177  df-xor 1305 This theorem is referenced by:  mtp-xor  1536
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