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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 2141 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
2 | 1 | notbid 624 | . 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶)) |
3 | df-nel 2340 | . 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶) | |
4 | df-nel 2340 | . 2 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
5 | 2, 3, 4 | 3bitr4g 221 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∉ wnel 2339 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-clel 2077 df-nel 2340 |
This theorem is referenced by: neleq12d 2345 ruALT 4294 |
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