Theorem not12an2impnot1 42141

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 42141 using completeusersproof, which is verified by the Metamath program. https://us.metamath.org/other/completeusersproof/not12an2impnot1ro.html 42141 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	⊢ ((¬ (𝜑 ∧ 𝜓) ∧ 𝜓) → ¬ 𝜑)

Proof of Theorem not12an2impnot1

Step	Hyp	Ref	Expression
1		pm3.21 471	. . 3 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	con3rr3 155	. 2 ⊢ (¬ (𝜑 ∧ 𝜓) → (𝜓 → ¬ 𝜑))
3	2	imp 406	1 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ 𝜓) → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  sineq0ALT  42510