![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > syl6eqelr | GIF version |
Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
syl6eqelr.1 | ⊢ (𝜑 → 𝐵 = 𝐴) |
syl6eqelr.2 | ⊢ 𝐵 ∈ 𝐶 |
Ref | Expression |
---|---|
syl6eqelr | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6eqelr.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) | |
2 | 1 | eqcomd 2094 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
3 | syl6eqelr.2 | . 2 ⊢ 𝐵 ∈ 𝐶 | |
4 | 2, 3 | syl6eqel 2179 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ∈ wcel 1439 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-4 1446 ax-17 1465 ax-ial 1473 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-cleq 2082 df-clel 2085 |
This theorem is referenced by: eusvnfb 4291 releldm2 5971 mapprc 6425 ixpprc 6492 ixpssmap2g 6500 ixpssmapg 6501 bren 6520 brdomg 6521 mapen 6618 ssenen 6623 ioof 9452 hashfacen 10304 fisum 10841 |
Copyright terms: Public domain | W3C validator |