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Mirrors > Home > NFE Home > Th. List > cbvopab1 | 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 1619 | . . . . 5 | |
2 | nfv 1619 | . . . . . . 7 | |
3 | nfs1v 2106 | . . . . . . 7 | |
4 | 2, 3 | nfan 1824 | . . . . . 6 |
5 | 4 | nfex 1843 | . . . . 5 |
6 | opeq1 4578 | . . . . . . . 8 | |
7 | 6 | eqeq2d 2364 | . . . . . . 7 |
8 | sbequ12 1919 | . . . . . . 7 | |
9 | 7, 8 | anbi12d 691 | . . . . . 6 |
10 | 9 | exbidv 1626 | . . . . 5 |
11 | 1, 5, 10 | cbvex 1985 | . . . 4 |
12 | nfv 1619 | . . . . . . 7 | |
13 | cbvopab1.1 | . . . . . . . 8 | |
14 | 13 | nfsb 2109 | . . . . . . 7 |
15 | 12, 14 | nfan 1824 | . . . . . 6 |
16 | 15 | nfex 1843 | . . . . 5 |
17 | nfv 1619 | . . . . 5 | |
18 | opeq1 4578 | . . . . . . . 8 | |
19 | 18 | eqeq2d 2364 | . . . . . . 7 |
20 | sbequ 2060 | . . . . . . . 8 | |
21 | cbvopab1.2 | . . . . . . . . 9 | |
22 | cbvopab1.3 | . . . . . . . . 9 | |
23 | 21, 22 | sbie 2038 | . . . . . . . 8 |
24 | 20, 23 | syl6bb 252 | . . . . . . 7 |
25 | 19, 24 | anbi12d 691 | . . . . . 6 |
26 | 25 | exbidv 1626 | . . . . 5 |
27 | 16, 17, 26 | cbvex 1985 | . . . 4 |
28 | 11, 27 | bitri 240 | . . 3 |
29 | 28 | abbii 2465 | . 2 |
30 | df-opab 4623 | . 2 | |
31 | df-opab 4623 | . 2 | |
32 | 29, 30, 31 | 3eqtr4i 2383 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wex 1541 wnf 1544 wceq 1642 wsb 1648 cab 2339 cop 4561 copab 4622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-opab 4623 |
This theorem is referenced by: cbvopab1v 4635 cbvmpt 5676 |
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