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| Description: A variation on impbid 212 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.) | 
| Ref | Expression | 
|---|---|
| impcon4bid.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) | 
| impcon4bid.2 | ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒)) | 
| Ref | Expression | 
|---|---|
| impcon4bid | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | impcon4bid.1 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
| 2 | impcon4bid.2 | . . 3 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒)) | |
| 3 | 2 | con4d 115 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) | 
| 4 | 1, 3 | impbid 212 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 | 
| 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 207 | 
| This theorem is referenced by: con4bid 317 soisoi 7348 isomin 7357 naddel1 8725 alephdom 10121 nn0n0n1ge2b 12595 om2uzlt2i 13992 sadcaddlem 16494 isprm5 16744 pcdvdsb 16907 om2noseqlt2 28306 expgt0b 32818 oexpreposd 42357 tfsconcatb0 43357 cvgdvgrat 44332 | 
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