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Theorem not12an2impnot1 41361
 Description: If a double conjunction is false and the second conjunct is true, then the first conjunct is false. https://us.metamath.org/other/completeusersproof/not12an2impnot1vd.html is the Virtual Deduction proof verified by automatically transforming it into the Metamath proof of not12an2impnot1 41361 using completeusersproof, which is verified by the Metamath program. https://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html 41361 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 pm3.21 475 . . 3 (𝜓 → (𝜑 → (𝜑𝜓)))
21con3rr3 158 . 2 (¬ (𝜑𝜓) → (𝜓 → ¬ 𝜑))
32imp 410 1 ((¬ (𝜑𝜓) ∧ 𝜓) → ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399 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 210  df-an 400 This theorem is referenced by:  sineq0ALT  41730
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