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Mirrors > Home > ILE Home > Th. List > anabs7 | GIF version |
Description: Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.) |
Ref | Expression |
---|---|
anabs7 | ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 108 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
2 | 1 | pm4.71ri 384 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ (𝜑 ∧ 𝜓))) |
3 | 2 | bicomi 130 | 1 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: (None) |
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