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Mirrors > Home > ILE Home > Th. List > sbcel21v | GIF version |
Description: Class substitution into a membership relation. One direction of sbcel2gv 3041 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Ref | Expression |
---|---|
sbcel21v | ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2986 | . 2 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐵 ∈ V) | |
2 | sbcel2gv 3041 | . . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | |
3 | 2 | biimpd 144 | . 2 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)) |
4 | 1, 3 | mpcom 36 | 1 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 Vcvv 2752 [wsbc 2977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-sbc 2978 |
This theorem is referenced by: (None) |
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