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Mirrors > Home > MPE Home > Th. List > impcon4bid | Structured version Visualization version GIF version |
Description: A variation on impbid 213 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 213 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 |
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 208 |
This theorem is referenced by: con4bid 318 soisoi 7070 isomin 7079 alephdom 9495 nn0n0n1ge2b 11951 om2uzlt2i 13307 sadcaddlem 15794 isprm5 16039 pcdvdsb 16193 oexpreposd 39057 cvgdvgrat 40522 |
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