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Mirrors > Home > ILE Home > Th. List > rbaib | GIF version |
Description: Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
baib.1 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
Ref | Expression |
---|---|
rbaib | ⊢ (𝜒 → (𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baib.1 | . . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) | |
2 | ancom 266 | . . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓)) | |
3 | 1, 2 | bitri 184 | . 2 ⊢ (𝜑 ↔ (𝜒 ∧ 𝜓)) |
4 | 3 | baib 919 | 1 ⊢ (𝜒 → (𝜑 ↔ 𝜓)) |
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: reusv1 4458 opres 4916 cores 5132 fvres 5539 fzsplit2 10049 cnptoprest 13709 |
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