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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-csbsn | Structured version Visualization version GIF version |
Description: Substitution in a singleton. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-csbsn | ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-csbsnlem 34117 | . . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦} | |
2 | 1 | csbeq2i 3888 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦} |
3 | csbcow 3895 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥} | |
4 | bj-csbsnlem 34117 | . 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴} | |
5 | 2, 3, 4 | 3eqtr3i 2849 | 1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⦋csb 3880 {csn 4557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-sbc 3770 df-csb 3881 df-sn 4558 |
This theorem is referenced by: bj-snsetex 34172 |
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