Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbel1 | Structured version Visualization version GIF version |
Description: Version of sbcel1g 4367 when substituting a set. (Note: one could have a corresponding version of sbcel12 4362 when substituting a set, but the point here is that the antecedent of sbcel1g 4367 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-sbel1 | ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbsbc 3778 | . 2 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝐵) | |
2 | sbcel1g 4367 | . . 3 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵)) | |
3 | 2 | elv 3501 | . 2 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵) |
4 | 1, 3 | bitri 277 | 1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 [wsb 2069 ∈ wcel 2114 Vcvv 3496 [wsbc 3774 ⦋csb 3885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-nul 4294 |
This theorem is referenced by: bj-snsetex 34277 |
Copyright terms: Public domain | W3C validator |