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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 524 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
2 | 1 | bicomd 212 | . 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓)) |
3 | 2 | pm5.32i 667 | 1 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 |
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 196 df-an 385 |
This theorem is referenced by: axrep5 4704 axsep2 4710 bj-axrep5 31980 elinintrab 36902 2sb5nd 37797 eelTT1 37956 uun121 38031 uunTT1 38041 uunTT1p1 38042 uunTT1p2 38043 uun111 38053 uun2221 38061 uun2221p1 38062 uun2221p2 38063 2sb5ndVD 38168 2sb5ndALT 38190 |
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