![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > neleq1 | GIF version |
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) |
Ref | Expression |
---|---|
neleq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2252 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | 1 | notbid 668 | . 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)) |
3 | df-nel 2456 | . 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶) | |
4 | df-nel 2456 | . 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
5 | 2, 3, 4 | 3bitr4g 223 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∉ wnel 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 df-clel 2185 df-nel 2456 |
This theorem is referenced by: neleq12d 2461 ruALT 4568 |
Copyright terms: Public domain | W3C validator |