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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbel1 | Structured version Visualization version GIF version |
Description: Version of sbcel1g 4421 when substituting a set. (Note: one could have a corresponding version of sbcel12 4416 when substituting a set, but the point here is that the antecedent of sbcel1g 4421 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 3795 | . 2 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝐵) | |
2 | sbcel1g 4421 | . . 3 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵)) | |
3 | 2 | elv 3482 | . 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 2060 ∈ wcel 2104 Vcvv 3477 [wsbc 3791 ⦋csb 3908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-nul 4340 |
This theorem is referenced by: bj-snsetex 36906 |
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