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Mirrors > Home > NFE 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 2413 | . 2 ⊢ (A = B → (A ∈ C ↔ B ∈ C)) | |
2 | eleq2 2414 | . 2 ⊢ (C = D → (B ∈ C ↔ B ∈ D)) | |
3 | 1, 2 | sylan9bb 680 | 1 ⊢ ((A = B ∧ C = D) → (A ∈ C ↔ B ∈ D)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-cleq 2346 df-clel 2349 |
This theorem is referenced by: nnsucelr 4428 ncfinlower 4483 sfindbl 4530 |
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