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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 34215 | . . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦} | |
2 | 1 | csbeq2i 3890 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦} |
3 | csbcow 3897 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥} | |
4 | bj-csbsnlem 34215 | . 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴} | |
5 | 2, 3, 4 | 3eqtr3i 2852 | 1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⦋csb 3882 {csn 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3772 df-csb 3883 df-sn 4561 |
This theorem is referenced by: bj-snsetex 34270 |
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