Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > abid2 | Structured version Visualization version GIF version |
Description: A simplification of class abstraction. Commuted form of abid1 2744. See comments there. (Contributed by NM, 26-Dec-1993.) |
Ref | Expression |
---|---|
abid2 | ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid1 2744 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
2 | 1 | eqcomi 2631 | 1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 {cab 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 |
This theorem is referenced by: csbid 3541 abss 3671 ssab 3672 abssi 3677 notab 3897 dfrab3 3902 notrab 3904 eusn 4265 uniintsn 4514 iunid 4575 csbexg 4792 imai 5478 dffv4 6188 orduniss2 7033 dfixp 7910 euen1b 8027 modom2 8162 infmap2 9040 cshwsexa 13570 ustfn 22005 ustn0 22024 fpwrelmap 29508 eulerpartlemgvv 30438 ballotlem2 30550 dffv5 32031 ptrest 33408 cnambfre 33458 cnvepresex 34104 pmapglb 35056 polval2N 35192 rngunsnply 37743 iocinico 37797 |
Copyright terms: Public domain | W3C validator |