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| Mirrors > Home > MPE Home > Th. List > mpbiran2d | Structured version Visualization version GIF version | ||
| Description: Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| mpbiran2d.1 | ⊢ (𝜑 → 𝜃) | 
| mpbiran2d.2 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | 
| Ref | Expression | 
|---|---|
| mpbiran2d | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mpbiran2d.1 | . 2 ⊢ (𝜑 → 𝜃) | |
| 2 | mpbiran2d.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
| 3 | 2 | biancomd 463 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∧ 𝜒))) | 
| 4 | 1, 3 | mpbirand 707 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 | 
| 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 df-an 396 | 
| This theorem is referenced by: opelidres 6009 funsnfsupp 9432 discld 23097 cncfcdm 24924 itgfsum 25862 dchreq 27302 lgsneg 27365 lgsquadlem2 27425 dfconngr1 30207 cover2 37722 iscnrm3rlem6 48842 0funcglem 48916 0funcg2 48917 thincmon 49082 thincepi 49083 | 
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