Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > impcon4bid | Structured version Visualization version GIF version |
Description: A variation on impbid 215 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 215 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 |
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 210 |
This theorem is referenced by: con4bid 320 soisoi 7137 isomin 7146 alephdom 9695 nn0n0n1ge2b 12158 om2uzlt2i 13524 sadcaddlem 16016 isprm5 16264 pcdvdsb 16422 naddel1 33576 oexpreposd 40028 cvgdvgrat 41604 |
Copyright terms: Public domain | W3C validator |