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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 2045 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
3 | syl6eqelr.2 | . 2 ⊢ 𝐵 ∈ 𝐶 | |
4 | 2, 3 | syl6eqel 2128 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 |
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-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 |
This theorem is referenced by: eusvnfb 4186 releldm2 5811 bren 6228 brdomg 6229 ioof 8840 |
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