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Mirrors > Home > ILE Home > Th. List > anabsi5 | GIF version |
Description: Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.) |
Ref | Expression |
---|---|
anabsi5.1 | ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒)) |
Ref | Expression |
---|---|
anabsi5 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anabsi5.1 | . . 3 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒)) | |
2 | 1 | imp 123 | . 2 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒) |
3 | 2 | anabss5 550 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: anabsi6 552 anabsi8 554 3anidm12 1256 equsexd 1690 rspce 2755 phplem3g 6703 ltexprlemrl 7366 ltexprlemru 7368 dvdssq 11565 |
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