Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > anabsan | Structured version Visualization version GIF version |
Description: Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) |
Ref | Expression |
---|---|
anabsan.1 | ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
anabsan | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.24 567 | . 2 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑)) | |
2 | anabsan.1 | . 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒) | |
3 | 1, 2 | sylanb 584 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 |
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 210 df-an 400 |
This theorem is referenced by: anabss1 666 anabss5 668 anandis 678 iddvds 15715 1dvds 15716 |
Copyright terms: Public domain | W3C validator |