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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 34344 | . . 3 ⊢ ⦋𝑦 / 𝑥⦌{𝑥} = {𝑦} | |
2 | 1 | csbeq2i 3836 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑦⦌{𝑦} |
3 | csbcow 3843 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌{𝑥} = ⦋𝐴 / 𝑥⦌{𝑥} | |
4 | bj-csbsnlem 34344 | . 2 ⊢ ⦋𝐴 / 𝑦⦌{𝑦} = {𝐴} | |
5 | 2, 3, 4 | 3eqtr3i 2829 | 1 ⊢ ⦋𝐴 / 𝑥⦌{𝑥} = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ⦋csb 3828 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-sbc 3721 df-csb 3829 df-sn 4526 |
This theorem is referenced by: bj-snsetex 34399 |
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