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Mirrors > Home > ILE Home > Th. List > sbcel2g | GIF version |
Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) |
Ref | Expression |
---|---|
sbcel2g | ⊢ (A ∈ 𝑉 → ([A / x]B ∈ 𝐶 ↔ B ∈ ⦋A / x⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcel12g 2859 | . 2 ⊢ (A ∈ 𝑉 → ([A / x]B ∈ 𝐶 ↔ ⦋A / x⦌B ∈ ⦋A / x⦌𝐶)) | |
2 | csbconstg 2858 | . . 3 ⊢ (A ∈ 𝑉 → ⦋A / x⦌B = B) | |
3 | 2 | eleq1d 2103 | . 2 ⊢ (A ∈ 𝑉 → (⦋A / x⦌B ∈ ⦋A / x⦌𝐶 ↔ B ∈ ⦋A / x⦌𝐶)) |
4 | 1, 3 | bitrd 177 | 1 ⊢ (A ∈ 𝑉 → ([A / x]B ∈ 𝐶 ↔ B ∈ ⦋A / x⦌𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∈ wcel 1390 [wsbc 2758 ⦋csb 2846 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-sbc 2759 df-csb 2847 |
This theorem is referenced by: csbcomg 2867 sbccsbg 2872 sbnfc2 2900 csbabg 2901 sbcssg 3324 csbunig 3579 csbxpg 4364 csbdmg 4472 csbrng 4725 bj-sels 9369 |
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