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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbel1 | Structured version Visualization version GIF version |
Description: Version of sbcel1g 3939 when substituting a set. (Note: one could have a corresponding version of sbcel12 3935 when substituting a set, but the point here is that the antecedent of sbcel1g 3939 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 3406 | . 2 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝐵) | |
2 | vex 3176 | . . 3 ⊢ 𝑦 ∈ V | |
3 | sbcel1g 3939 | . . 3 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵) |
5 | 1, 4 | bitri 263 | 1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 [wsb 1867 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 ⦋csb 3499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-nul 3875 |
This theorem is referenced by: bj-snsetex 32144 |
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