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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 8784 | . 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 1540 [cec 8743 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 |
| This theorem is referenced by: brecop 8850 eroveu 8852 erov 8854 ecovcom 8863 ecovass 8864 ecovdi 8865 addsrmo 11113 mulsrmo 11114 addsrpr 11115 mulsrpr 11116 supsrlem 11151 supsr 11152 qus0 19207 qusinv 19208 qussub 19209 sylow2blem2 19639 frgpadd 19781 vrgpval 19785 vrgpinv 19787 frgpup3lem 19795 qusabl 19883 quscrng 21293 pzriprnglem11 21502 pzriprnglem12 21503 qustgplem 24129 pi1addval 25081 pi1xfrf 25086 pi1xfrval 25087 pi1xfrcnvlem 25089 pi1xfrcnv 25090 pi1cof 25092 pi1coval 25093 pi1coghm 25094 vitalilem3 25645 elrlocbasi 33270 rlocaddval 33272 rlocmulval 33273 rloccring 33274 rloc0g 33275 rloc1r 33276 rlocf1 33277 idomsubr 33311 opprqusmulr 33519 zringfrac 33582 ismntoplly 34026 linedegen 36144 fvline 36145 aks5lem3a 42190 aks5lem5a 42192 aks5lem6 42193 |
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