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Mirrors > Home > ILE Home > Th. List > anidmdbi | GIF version |
Description: Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.) |
Ref | Expression |
---|---|
anidmdbi | ⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anidm 396 | . 2 ⊢ ((𝜓 ∧ 𝜓) ↔ 𝜓) | |
2 | 1 | imbi2i 226 | 1 ⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (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: (None) |
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