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Mirrors > Home > MPE Home > Th. List > anidmdbi | Structured version Visualization version 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 568 | . 2 ⊢ ((𝜓 ∧ 𝜓) ↔ 𝜓) | |
2 | 1 | imbi2i 339 | 1 ⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 |
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 210 df-an 400 |
This theorem is referenced by: nanim 1493 |
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