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Mirrors > Home > ILE Home > Th. List > anabs5 | 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 285 | . . 3 ⊢ (φ → (ψ ↔ (φ ∧ ψ))) | |
2 | 1 | bicomd 129 | . 2 ⊢ (φ → ((φ ∧ ψ) ↔ ψ)) |
3 | 2 | pm5.32i 427 | 1 ⊢ ((φ ∧ (φ ∧ ψ)) ↔ (φ ∧ ψ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: mo3h 1950 indif 3174 axsep2 3867 |
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