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Mirrors > Home > ILE Home > Th. List > 3anidm12 | GIF version |
Description: Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
Ref | Expression |
---|---|
3anidm12.1 | ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
3anidm12 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anidm12.1 | . . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | 3expib 1201 | . 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒)) |
3 | 2 | anabsi5 574 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 975 |
This theorem is referenced by: 3anidm13 1291 syl2an3an 1293 prarloclemarch2 7381 nq02m 7427 recexprlem1ssl 7595 recexprlem1ssu 7596 nncan 8148 dividap 8618 modqid0 10306 subsq 10582 retanclap 11685 tannegap 11691 gcd0id 11934 coprm 12098 |
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