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Mirrors > Home > NFE Home > Th. List > clos1eq1 | Unicode version |
Description: Equality law for closure. (Contributed by SF, 11-Feb-2015.) |
Ref | Expression |
---|---|
clos1eq1 | Clos1 Clos1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3292 | . . . . 5 | |
2 | 1 | anbi1d 685 | . . . 4 |
3 | 2 | abbidv 2467 | . . 3 |
4 | inteq 3929 | . . 3 | |
5 | 3, 4 | syl 15 | . 2 |
6 | df-clos1 5873 | . 2 Clos1 | |
7 | df-clos1 5873 | . 2 Clos1 | |
8 | 5, 6, 7 | 3eqtr4g 2410 | 1 Clos1 Clos1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 wceq 1642 cab 2339 wss 3257 cint 3926 cima 4722 Clos1 cclos1 5872 |
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 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 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-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-int 3927 df-clos1 5873 |
This theorem is referenced by: clos1exg 5877 clos1basesucg 5884 spacval 6282 nchoicelem11 6299 nchoicelem16 6304 freceq12 6311 frecxp 6314 |
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