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| Mirrors > Home > MPE Home > Th. List > sbcel2gv | Structured version Visualization version GIF version | ||
| Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
| Ref | Expression |
|---|---|
| sbcel2gv | ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq2 2830 | . 2 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) | |
| 2 | eleq2 2830 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
| 3 | 1, 2 | sbcie2g 3829 | 1 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 [wsbc 3788 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 |
| This theorem is referenced by: sbcel21v 3858 csbvarg 4434 bnj92 34876 bnj539 34905 minregex 43547 frege77 43953 sbcoreleleq 44555 trsbc 44560 onfrALTlem5 44562 sbcoreleleqVD 44879 trsbcVD 44897 onfrALTlem5VD 44905 modelaxreplem3 44997 |
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