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Mirrors > Home > MPE Home > Th. List > impcon4bid | Structured version Visualization version GIF version |
Description: A variation on impbid 211 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 211 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ 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: con4bid 316 soisoi 7255 isomin 7264 alephdom 9938 nn0n0n1ge2b 12402 om2uzlt2i 13772 sadcaddlem 16263 isprm5 16509 pcdvdsb 16667 naddel1 34118 oexpreposd 40589 cvgdvgrat 42260 |
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