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Mirrors > Home > MPE Home > Th. List > anabss7 | Structured version Visualization version GIF version |
Description: Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.) |
Ref | Expression |
---|---|
anabss7.1 | ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) → 𝜒) |
Ref | Expression |
---|---|
anabss7 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anabss7.1 | . . 3 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) → 𝜒) | |
2 | 1 | anassrs 471 | . 2 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜓) → 𝜒) |
3 | 2 | anabss4 667 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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: anabsan2 674 syl2an23an 1425 funbrfv 6763 faclbnd5 13864 lcmcllem 16153 funbrafv2 44411 |
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