Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > bibif | Structured version Visualization version GIF version |
Description: Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.) |
Ref | Expression |
---|---|
bibif | ⊢ (¬ 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbn2 370 | . 2 ⊢ (¬ 𝜓 → (¬ 𝜑 ↔ (𝜓 ↔ 𝜑))) | |
2 | bicom 221 | . 2 ⊢ ((𝜓 ↔ 𝜑) ↔ (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | bitr2di 287 | 1 ⊢ (¬ 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 |
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 |
This theorem is referenced by: nbn 372 bj-bibibi 34796 or3or 41655 |
Copyright terms: Public domain | W3C validator |