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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 376 | . 2 ⊢ ((𝜓 ∧ 𝜓) ↔ 𝜓) | |
2 | 1 | imbi2i 215 | 1 ⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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