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Mirrors > Home > MPE Home > Th. List > eceq1d | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.) |
Ref | Expression |
---|---|
eceq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
eceq1d | ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eceq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | eceq1 8769 | . 2 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 [cec 8729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ec 8733 |
This theorem is referenced by: brecop 8835 eroveu 8837 erov 8839 ecovcom 8848 ecovass 8849 ecovdi 8850 addsrmo 11104 mulsrmo 11105 addsrpr 11106 mulsrpr 11107 supsrlem 11142 supsr 11143 qus0 19151 qusinv 19152 qussub 19153 sylow2blem2 19583 frgpadd 19725 vrgpval 19729 vrgpinv 19731 frgpup3lem 19739 qusabl 19827 quscrng 21182 pzriprnglem11 21424 pzriprnglem12 21425 qustgplem 24045 pi1addval 24995 pi1xfrf 25000 pi1xfrval 25001 pi1xfrcnvlem 25003 pi1xfrcnv 25004 pi1cof 25006 pi1coval 25007 pi1coghm 25008 vitalilem3 25559 elrlocbasi 33005 rlocaddval 33007 rlocmulval 33008 rloccring 33009 rloc0g 33010 rloc1r 33011 rlocf1 33012 idomsubr 33020 opprqusmulr 33227 zringfrac 33277 ismntoplly 33659 linedegen 35772 fvline 35773 |
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