Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ibd | Structured version Visualization version GIF version |
Description: Deduction that converts a biconditional implied by one of its arguments, into an implication. Deduction associated with ibi 266. (Contributed by NM, 26-Jun-2004.) |
Ref | Expression |
---|---|
ibd.1 | ⊢ (𝜑 → (𝜓 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
ibd | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibd.1 | . 2 ⊢ (𝜑 → (𝜓 → (𝜓 ↔ 𝜒))) | |
2 | biimp 214 | . 2 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)) | |
3 | 1, 2 | syli 39 | 1 ⊢ (𝜑 → (𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → 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: sssn 4765 unblem2 9045 atcv0eq 30737 atcv1 30738 atomli 30740 atcvatlem 30743 ibdr 36868 |
Copyright terms: Public domain | W3C validator |