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Mirrors > Home > MPE Home > Th. List > anabs5 | Structured version Visualization version GIF version |
Description: Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) |
Ref | Expression |
---|---|
anabs5 | ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibar 527 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
2 | 1 | bicomd 222 | . 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓)) |
3 | 2 | pm5.32i 573 | 1 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 |
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 206 df-an 395 |
This theorem is referenced by: rmoanidOLD 3386 reuanidOLD 3387 axrep5 5290 elinintrab 42630 2sb5nd 43623 eelTT1 43773 uun121 43846 uunTT1 43856 uunTT1p1 43857 uunTT1p2 43858 uun111 43868 uun2221 43876 uun2221p1 43877 uun2221p2 43878 2sb5ndVD 43973 2sb5ndALT 43995 |
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