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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 466 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜃 ∧ 𝜒))) |
4 | 1, 3 | mpbirand 705 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 |
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 209 df-an 399 |
This theorem is referenced by: opelidres 5867 funsnfsupp 8859 discld 21699 cncffvrn 23508 itgfsum 24429 dchreq 25836 lgsneg 25899 lgsquadlem2 25959 dfconngr1 27969 cover2 34991 |
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