Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 3anidm13 | Structured version Visualization version GIF version |
Description: Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
Ref | Expression |
---|---|
3anidm13.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒) |
Ref | Expression |
---|---|
3anidm13 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anidm13.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒) | |
2 | 1 | 3com23 1122 | . 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒) |
3 | 2 | 3anidm12 1415 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
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: npncan2 10915 ltsubpos 11134 leaddle0 11157 subge02 11158 halfaddsub 11873 avglt1 11878 hashssdif 13776 pythagtriplem4 16158 pythagtriplem14 16167 lsmss2 18795 grpoidinvlem2 28284 hvpncan3 28821 bcm1n 30520 revpfxsfxrev 32364 resubidaddid1 39232 3anidm12p1 41147 3impcombi 41158 |
Copyright terms: Public domain | W3C validator |