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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 318 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | id 22 | . . 3 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓)) | |
4 | 3 | con4bid 317 | . 2 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → (𝜑 ↔ 𝜓)) |
5 | 2, 4 | impbii 208 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ 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: notbii 320 con4bii 321 con2bi 354 nbn2 371 pm5.32 575 norass 1539 hadnot 1604 had0 1606 cbvexdw 2336 cbvexd 2408 rexbiOLD 3106 rexprg 4700 isocnv3 7326 suppimacnv 8156 sumodd 16328 f1omvdco3 19312 ist0cld 32802 onsuct0 35315 bj-cbvexdv 35667 wl-3xornot 36351 ifpbi1 42214 ifpbi13 42226 abciffcbatnabciffncba 45626 abciffcbatnabciffncbai 45627 ichn 46111 |
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