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Mirrors > Home > ILE 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 2227 | . 2 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
2 | 1 | biimprcd 159 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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: elex22 2739 elex2 2740 reu6 2913 disjne 3460 ssimaex 5544 fnex 5704 f1ocnv2d 6039 tfrlem8 6280 eroprf 6588 ac6sfi 6858 recclnq 7327 prnmaddl 7425 renegcl 8153 nn0ind-raph 9302 iccid 9855 4sqlem1 12312 4sqlem4 12316 opnneiid 12762 metrest 13104 coseq0negpitopi 13355 bj-nn0suc 13739 bj-inf2vnlem2 13746 bj-nn0sucALT 13753 |
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