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| Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2115). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) | 
| Ref | Expression | 
|---|---|
| clelsb1 | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eleq1w 2823 | . 2 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) | |
| 2 | eleq1w 2823 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 3 | 1, 2 | sbievw2 2097 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 [wsb 2063 ∈ wcel 2107 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 df-clel 2815 | 
| This theorem is referenced by: hblem 2872 hblemg 2873 eqabdv 2874 clelsb1fw 2908 clelsb1f 2909 cbvreuwOLD 3414 cbvreu 3427 elrabi 3686 sbcel1v 3855 rmo3 3888 kmlem15 10206 iuninc 32574 measiuns 34219 ballotlemodife 34501 bj-nfcf 36925 ellimcabssub0 45637 | 
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