Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rabxfrd | Structured version Visualization version Unicode version |
Description: Class builder membership after substituting an expression (containing ) for in the class expression . (Contributed by NM, 16-Jan-2012.) |
Ref | Expression |
---|---|
rabxfrd.1 | |
rabxfrd.2 | |
rabxfrd.3 | |
rabxfrd.4 | |
rabxfrd.5 |
Ref | Expression |
---|---|
rabxfrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabxfrd.3 | . . . . . . . . . . 11 | |
2 | 1 | ex 450 | . . . . . . . . . 10 |
3 | ibibr 358 | . . . . . . . . . 10 | |
4 | 2, 3 | sylib 208 | . . . . . . . . 9 |
5 | 4 | imp 445 | . . . . . . . 8 |
6 | 5 | anbi1d 741 | . . . . . . 7 |
7 | rabxfrd.4 | . . . . . . . 8 | |
8 | 7 | elrab 3363 | . . . . . . 7 |
9 | rabid 3116 | . . . . . . 7 | |
10 | 6, 8, 9 | 3bitr4g 303 | . . . . . 6 |
11 | 10 | rabbidva 3188 | . . . . 5 |
12 | 11 | eleq2d 2687 | . . . 4 |
13 | rabxfrd.1 | . . . . 5 | |
14 | nfcv 2764 | . . . . 5 | |
15 | rabxfrd.2 | . . . . . 6 | |
16 | 15 | nfel1 2779 | . . . . 5 |
17 | rabxfrd.5 | . . . . . 6 | |
18 | 17 | eleq1d 2686 | . . . . 5 |
19 | 13, 14, 16, 18 | elrabf 3360 | . . . 4 |
20 | nfrab1 3122 | . . . . . 6 | |
21 | 13, 20 | nfel 2777 | . . . . 5 |
22 | eleq1 2689 | . . . . 5 | |
23 | 13, 14, 21, 22 | elrabf 3360 | . . . 4 |
24 | 12, 19, 23 | 3bitr3g 302 | . . 3 |
25 | pm5.32 668 | . . 3 | |
26 | 24, 25 | sylibr 224 | . 2 |
27 | 26 | imp 445 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wnfc 2751 crab 2916 |
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 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 |
This theorem is referenced by: rabxfr 4890 riotaxfrd 6642 |
Copyright terms: Public domain | W3C validator |