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Mirrors > Home > ILE Home > Th. List > eleq12 | GIF version |
Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
Ref | Expression |
---|---|
eleq12 | ⊢ ((A = B ∧ 𝐶 = 𝐷) → (A ∈ 𝐶 ↔ B ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2097 | . 2 ⊢ (A = B → (A ∈ 𝐶 ↔ B ∈ 𝐶)) | |
2 | eleq2 2098 | . 2 ⊢ (𝐶 = 𝐷 → (B ∈ 𝐶 ↔ B ∈ 𝐷)) | |
3 | 1, 2 | sylan9bb 435 | 1 ⊢ ((A = B ∧ 𝐶 = 𝐷) → (A ∈ 𝐶 ↔ B ∈ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ 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: trel 3852 pwnss 3903 epelg 4018 preleq 4233 acexmid 5454 |
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