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Mirrors > Home > ILE Home > Th. List > eleq12i | GIF version |
Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
Ref | Expression |
---|---|
eleq1i.1 | ⊢ A = B |
eleq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
eleq12i | ⊢ (A ∈ 𝐶 ↔ B ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
2 | 1 | eleq2i 2101 | . 2 ⊢ (A ∈ 𝐶 ↔ A ∈ 𝐷) |
3 | eleq1i.1 | . . 3 ⊢ A = B | |
4 | 3 | eleq1i 2100 | . 2 ⊢ (A ∈ 𝐷 ↔ B ∈ 𝐷) |
5 | 2, 4 | bitri 173 | 1 ⊢ (A ∈ 𝐶 ↔ B ∈ 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1242 ∈ wcel 1390 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-cleq 2030 df-clel 2033 |
This theorem is referenced by: 3eltr3g 2119 3eltr4g 2120 sbcel12g 2859 |
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