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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 32090 | . . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦} | |
2 | 1 | csbeq2i 3945 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦} |
3 | csbco 3509 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥} | |
4 | bj-csbsnlem 32090 | . 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴} | |
5 | 2, 3, 4 | 3eqtr3i 2640 | 1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⦋csb 3499 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 df-sn 4126 |
This theorem is referenced by: bj-snsetex 32144 |
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