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Mirrors > Home > NFE Home > Th. List > eleq1a | GIF version |
Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.) |
Ref | Expression |
---|---|
eleq1a | ⊢ (A ∈ B → (C = A → C ∈ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2413 | . 2 ⊢ (C = A → (C ∈ B ↔ A ∈ B)) | |
2 | 1 | biimprcd 216 | 1 ⊢ (A ∈ B → (C = A → C ∈ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: elex22 2870 elex2 2871 reu6 3025 disjne 3596 evennn 4506 oddnn 4507 nnadjoin 4520 phi11lem1 4595 f1o2d 5727 |
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