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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 8782 | . 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 1536 [cec 8741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ec 8745 |
This theorem is referenced by: brecop 8848 eroveu 8850 erov 8852 ecovcom 8861 ecovass 8862 ecovdi 8863 addsrmo 11110 mulsrmo 11111 addsrpr 11112 mulsrpr 11113 supsrlem 11148 supsr 11149 qus0 19219 qusinv 19220 qussub 19221 sylow2blem2 19653 frgpadd 19795 vrgpval 19799 vrgpinv 19801 frgpup3lem 19809 qusabl 19897 quscrng 21310 pzriprnglem11 21519 pzriprnglem12 21520 qustgplem 24144 pi1addval 25094 pi1xfrf 25099 pi1xfrval 25100 pi1xfrcnvlem 25102 pi1xfrcnv 25103 pi1cof 25105 pi1coval 25106 pi1coghm 25107 vitalilem3 25658 elrlocbasi 33252 rlocaddval 33254 rlocmulval 33255 rloccring 33256 rloc0g 33257 rloc1r 33258 rlocf1 33259 idomsubr 33290 opprqusmulr 33498 zringfrac 33561 ismntoplly 33987 linedegen 36124 fvline 36125 aks5lem3a 42170 aks5lem5a 42172 aks5lem6 42173 |
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