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Mirrors > Home > MPE Home > Th. List > anabss1 | Structured version Visualization version GIF version |
Description: Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.) |
Ref | Expression |
---|---|
anabss1.1 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒) |
Ref | Expression |
---|---|
anabss1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anabss1.1 | . . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒) | |
2 | 1 | an32s 846 | . 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒) |
3 | 2 | anabsan 854 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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: anabss4 856 ordtri3or 5755 onfununi 7438 omordi 7646 oeoelem 7678 hashssdif 13200 nzss 38516 stirlinglem5 40295 |
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