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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 525 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
2 | 1 | bicomd 213 | . 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓)) |
3 | 2 | pm5.32i 669 | 1 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 |
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 197 df-an 386 |
This theorem is referenced by: axrep5 4776 axsep2 4782 bj-axrep5 32792 elinintrab 37883 2sb5nd 38776 eelTT1 38935 uun121 39010 uunTT1 39020 uunTT1p1 39021 uunTT1p2 39022 uun111 39032 uun2221 39040 uun2221p1 39041 uun2221p2 39042 2sb5ndVD 39146 2sb5ndALT 39168 |
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