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Theorem not12an2impnot1 38305
Description: If a double conjunction is false and the second conjunct is true, then the first conjunct is false. http://us.metamath.org/other/completeusersproof/not12an2impnot1vd.html is the Virtual Deduction proof verified by automatically transforming it into the Metamath proof of not12an2impnot1 38305 using completeusersproof, which is verified by the Metamath program. http://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.)
Assertion
Ref Expression
not12an2impnot1 ((¬ (𝜑𝜓) ∧ 𝜓) → ¬ 𝜑)

Proof of Theorem not12an2impnot1
StepHypRef Expression
1 pm3.21 464 . . 3 (𝜓 → (𝜑 → (𝜑𝜓)))
21con3rr3 151 . 2 (¬ (𝜑𝜓) → (𝜓 → ¬ 𝜑))
32imp 445 1 ((¬ (𝜑𝜓) ∧ 𝜓) → ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384
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 197  df-an 386
This theorem is referenced by:  sineq0ALT  38695
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