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Mirrors > Home > MPE Home > Th. List > cbvopab1 | Structured version Visualization version Unicode version |
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
cbvopab1.1 | |
cbvopab1.2 | |
cbvopab1.3 |
Ref | Expression |
---|---|
cbvopab1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . . . . 5 | |
2 | nfv 1843 | . . . . . . 7 | |
3 | nfs1v 2437 | . . . . . . 7 | |
4 | 2, 3 | nfan 1828 | . . . . . 6 |
5 | 4 | nfex 2154 | . . . . 5 |
6 | opeq1 4402 | . . . . . . . 8 | |
7 | 6 | eqeq2d 2632 | . . . . . . 7 |
8 | sbequ12 2111 | . . . . . . 7 | |
9 | 7, 8 | anbi12d 747 | . . . . . 6 |
10 | 9 | exbidv 1850 | . . . . 5 |
11 | 1, 5, 10 | cbvex 2272 | . . . 4 |
12 | nfv 1843 | . . . . . . 7 | |
13 | cbvopab1.1 | . . . . . . . 8 | |
14 | 13 | nfsb 2440 | . . . . . . 7 |
15 | 12, 14 | nfan 1828 | . . . . . 6 |
16 | 15 | nfex 2154 | . . . . 5 |
17 | nfv 1843 | . . . . 5 | |
18 | opeq1 4402 | . . . . . . . 8 | |
19 | 18 | eqeq2d 2632 | . . . . . . 7 |
20 | sbequ 2376 | . . . . . . . 8 | |
21 | cbvopab1.2 | . . . . . . . . 9 | |
22 | cbvopab1.3 | . . . . . . . . 9 | |
23 | 21, 22 | sbie 2408 | . . . . . . . 8 |
24 | 20, 23 | syl6bb 276 | . . . . . . 7 |
25 | 19, 24 | anbi12d 747 | . . . . . 6 |
26 | 25 | exbidv 1850 | . . . . 5 |
27 | 16, 17, 26 | cbvex 2272 | . . . 4 |
28 | 11, 27 | bitri 264 | . . 3 |
29 | 28 | abbii 2739 | . 2 |
30 | df-opab 4713 | . 2 | |
31 | df-opab 4713 | . 2 | |
32 | 29, 30, 31 | 3eqtr4i 2654 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wex 1704 wnf 1708 wsb 1880 cab 2608 cop 4183 copab 4712 |
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-3an 1039 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-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 |
This theorem is referenced by: cbvopab1v 4726 cbvmptf 4748 cbvmpt 4749 phpreu 33393 poimirlem26 33435 mbfposadd 33457 |
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