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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 4701 isocnv3 7329 suppimacnv 8159 sumodd 16331 f1omvdco3 19317 ist0cld 32813 onsuct0 35326 bj-cbvexdv 35678 wl-3xornot 36362 ifpbi1 42228 ifpbi13 42240 abciffcbatnabciffncba 45639 abciffcbatnabciffncbai 45640 ichn 46124 |
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