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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbel1 | Structured version Visualization version GIF version |
Description: Version of sbcel1g 3987 when substituting a set. (Note: one could have a corresponding version of sbcel12 3983 when substituting a set, but the point here is that the antecedent of sbcel1g 3987 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 3439 | . 2 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝐵) | |
2 | vex 3203 | . . 3 ⊢ 𝑦 ∈ V | |
3 | sbcel1g 3987 | . . 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 264 | 1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 [wsb 1880 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 ⦋csb 3533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-nul 3916 |
This theorem is referenced by: bj-snsetex 32951 |
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