Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > neleq2 | GIF version |
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) |
Ref | Expression |
---|---|
neleq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2101 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | |
2 | 1 | notbid 592 | . 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 ∈ 𝐴 ↔ ¬ 𝐶 ∈ 𝐵)) |
3 | df-nel 2207 | . 2 ⊢ (𝐶 ∉ 𝐴 ↔ ¬ 𝐶 ∈ 𝐴) | |
4 | df-nel 2207 | . 2 ⊢ (𝐶 ∉ 𝐵 ↔ ¬ 𝐶 ∈ 𝐵) | |
5 | 2, 3, 4 | 3bitr4g 212 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∉ wnel 2205 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-clel 2036 df-nel 2207 |
This theorem is referenced by: neleq12d 2303 |
Copyright terms: Public domain | W3C validator |