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Mirrors > Home > MPE Home > Th. List > 3biant1d | Structured version Visualization version GIF version |
Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 528. (Contributed by Alexander van der Vekens, 26-Sep-2017.) |
Ref | Expression |
---|---|
3biantd.1 | ⊢ (𝜑 → 𝜃) |
Ref | Expression |
---|---|
3biant1d | ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3biantd.1 | . . 3 ⊢ (𝜑 → 𝜃) | |
2 | 1 | biantrurd 529 | . 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ (𝜒 ∧ 𝜓)))) |
3 | 3anass 1042 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜓) ↔ (𝜃 ∧ (𝜒 ∧ 𝜓))) | |
4 | 2, 3 | syl6bbr 278 | 1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 |
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 197 df-an 386 df-3an 1039 |
This theorem is referenced by: metuel2 22370 itgsubst 23812 clwlkclwwlk 26903 dfgcd3 33170 itg2addnclem2 33462 |
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