Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > simpr3l | Structured version Visualization version GIF version |
Description: Simplification of conjunction. (Contributed by NM, 9-Mar-2012.) (Proof shortened by Wolf Lammen, 24-Jun-2022.) |
Ref | Expression |
---|---|
simpr3l | ⊢ ((𝜏 ∧ (𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓))) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 769 | . 2 ⊢ ((𝜏 ∧ (𝜑 ∧ 𝜓)) → 𝜑) | |
2 | 1 | 3ad2antr3 1186 | 1 ⊢ ((𝜏 ∧ (𝜒 ∧ 𝜃 ∧ (𝜑 ∧ 𝜓))) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 |
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 209 df-an 399 df-3an 1085 |
This theorem is referenced by: ax5seg 26726 axcont 26764 nosupbnd1lem5 33214 segconeq 33473 idinside 33547 btwnconn1lem10 33559 segletr 33577 cdlemc3 37331 cdlemc4 37332 cdleme1 37365 cdleme2 37366 cdleme3b 37367 cdleme3c 37368 cdleme3e 37370 cdleme27a 37505 stoweidlem56 42348 |
Copyright terms: Public domain | W3C validator |