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Mirrors > Home > ILE Home > Th. List > baibd | GIF version |
Description: Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
baibd.1 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
Ref | Expression |
---|---|
baibd | ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baibd.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
2 | ibar 301 | . . 3 ⊢ (𝜒 → (𝜃 ↔ (𝜒 ∧ 𝜃))) | |
3 | 2 | bicomd 141 | . 2 ⊢ (𝜒 → ((𝜒 ∧ 𝜃) ↔ 𝜃)) |
4 | 1, 3 | sylan9bb 462 | 1 ⊢ ((𝜑 ∧ 𝜒) → (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: eluz 9527 elicc4 9924 divalgmodcl 11913 iscrng2 13021 iscld2 13264 cncnp2m 13391 cnnei 13392 reopnap 13698 cnlimc 13801 |
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