![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > notbi | Structured version Visualization version GIF version |
Description: Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.) |
Ref | Expression |
---|---|
notbi | ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓)) | |
2 | 1 | notbid 307 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | id 22 | . . 3 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓)) | |
4 | 3 | con4bid 306 | . 2 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (𝜑 ↔ 𝜓)) |
5 | 2, 4 | impbii 199 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 |
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 |
This theorem is referenced by: notbii 309 con4bii 310 con2bi 342 nbn2 359 pm5.32 557 hadnot 1689 had0 1691 cbvexd 2437 symdifass 4003 isocnv3 6726 suppimacnv 7458 sumodd 15320 f1omvdco3 18077 onsuct0 32778 bj-cbvexdv 33073 ifpbi1 38349 ifpbi13 38361 abciffcbatnabciffncba 41617 abciffcbatnabciffncbai 41618 |
Copyright terms: Public domain | W3C validator |