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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbel1 | Structured version Visualization version GIF version | ||
| Description: Version of sbcel1g 4415 when substituting a set. (Note: one could have a corresponding version of sbcel12 4410 when substituting a set, but the point here is that the antecedent of sbcel1g 4415 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 3791 | . 2 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝐵) | |
| 2 | sbcel1g 4415 | . . 3 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵)) | |
| 3 | 2 | elv 3484 | . 2 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵) |
| 4 | 1, 3 | bitri 275 | 1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 [wsb 2064 ∈ wcel 2108 Vcvv 3479 [wsbc 3787 ⦋csb 3898 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-nul 4333 |
| This theorem is referenced by: bj-snsetex 36942 |
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