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Mirrors > Home > ILE Home > Th. List > mpbirand | GIF version |
Description: Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
mpbirand.1 | ⊢ (𝜑 → 𝜒) |
mpbirand.2 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
Ref | Expression |
---|---|
mpbirand | ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpbirand.2 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
2 | mpbirand.1 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | 2 | biantrurd 305 | . 2 ⊢ (𝜑 → (𝜃 ↔ (𝜒 ∧ 𝜃))) |
4 | 1, 3 | bitr4d 191 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: fprodssdc 11593 dvdsr2d 13217 txmetcn 13912 |
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