Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 648 | . 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒) |
3 | 2 | anabsan 661 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 |
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 206 df-an 396 |
This theorem is referenced by: anabss4 663 ordtri3or 6295 onfununi 8156 omordi 8373 oeoelem 8405 hashssdif 14108 fzindd 39964 nzss 41888 stirlinglem5 43573 |
Copyright terms: Public domain | W3C validator |