Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > not12an2impnot1 | Structured version Visualization version GIF version |
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 38784 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.) |
Ref | Expression |
---|---|
not12an2impnot1 | ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ 𝜓) → ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.21 464 | . . 3 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
2 | 1 | con3rr3 151 | . 2 ⊢ (¬ (𝜑 ∧ 𝜓) → (𝜓 → ¬ 𝜑)) |
3 | 2 | imp 445 | 1 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ 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 39173 |
Copyright terms: Public domain | W3C validator |