![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > bianfi | Structured version Visualization version GIF version |
Description: A wff conjoined with falsehood is false. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) |
Ref | Expression |
---|---|
bianfi.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
bianfi | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bianfi.1 | . 2 ⊢ ¬ 𝜑 | |
2 | 1 | intnan 951 | . 2 ⊢ ¬ (𝜓 ∧ 𝜑) |
3 | 1, 2 | 2false 364 | 1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 |
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 196 df-an 385 |
This theorem is referenced by: in0 3920 opthprc 5089 ind1a 29410 |
Copyright terms: Public domain | W3C validator |