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Mirrors > Home > MPE Home > Th. List > elabgf | Unicode version |
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
elabgf.1 | |
elabgf.2 | |
elabgf.3 |
Ref | Expression |
---|---|
elabgf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabgf.1 | . 2 | |
2 | nfab1 2621 | . . . 4 | |
3 | 1, 2 | nfel 2632 | . . 3 |
4 | elabgf.2 | . . 3 | |
5 | 3, 4 | nfbi 1934 | . 2 |
6 | eleq1 2529 | . . 3 | |
7 | elabgf.3 | . . 3 | |
8 | 6, 7 | bibi12d 321 | . 2 |
9 | abid 2444 | . 2 | |
10 | 1, 5, 8, 9 | vtoclgf 3165 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
= wceq 1395 F/ wnf 1616 e. wcel 1818
{ cab 2442 F/_ wnfc 2605 |
This theorem is referenced by: elabf 3245 elabg 3247 elab3gf 3251 elrabf 3255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 |
This theorem depends on definitions: df-bi 185 df-an 371 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-v 3111 |
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