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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 2231 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷) |
3 | eleq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
4 | 3 | eleq1i 2230 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷) |
5 | 2, 4 | bitri 183 | 1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1342 ∈ wcel 2135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-cleq 2157 df-clel 2160 |
This theorem is referenced by: 3eltr3g 2249 3eltr4g 2250 sbcel12g 3055 ennnfonelem1 12277 |
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