| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > sbcel21v | Structured version Visualization version GIF version | ||
| Description: Class substitution into a membership relation. One direction of sbcel2gv 3857 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 3798 | . 2 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐵 ∈ V) | |
| 2 | sbcel2gv 3857 | . . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | |
| 3 | 2 | biimpd 229 | . 2 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)) |
| 4 | 1, 3 | mpcom 38 | 1 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 [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-v 3482 df-sbc 3789 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |