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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 465 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∧ 𝜒))) |
4 | 1, 3 | mpbirand 706 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 |
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 df-an 398 |
This theorem is referenced by: opelidres 5988 funsnfsupp 9375 discld 22562 cncfcdm 24383 itgfsum 25313 dchreq 26728 lgsneg 26791 lgsquadlem2 26851 dfconngr1 29408 cover2 36488 iscnrm3rlem6 47418 thincmon 47494 thincepi 47495 |
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