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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 36882 | . . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦} | |
| 2 | 1 | csbeq2i 3906 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦} |
| 3 | csbcow 3913 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥} | |
| 4 | bj-csbsnlem 36882 | . 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴} | |
| 5 | 2, 3, 4 | 3eqtr3i 2772 | 1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⦋csb 3898 {csn 4624 |
| 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-12 2177 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-sbc 3788 df-csb 3899 df-sn 4625 |
| This theorem is referenced by: bj-snsetex 36942 |
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