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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 7951 | . 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 1632 [cec 7911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-xp 5272 df-cnv 5274 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-ec 7915 |
This theorem is referenced by: brecop 8009 eroveu 8011 erov 8013 ecovcom 8022 ecovass 8023 ecovdi 8024 addsrmo 10106 mulsrmo 10107 addsrpr 10108 mulsrpr 10109 supsrlem 10144 supsr 10145 qus0 17873 qusinv 17874 qussub 17875 sylow2blem2 18256 frgpadd 18396 vrgpval 18400 vrgpinv 18402 frgpup3lem 18410 qusabl 18488 quscrng 19462 qustgplem 22145 pi1addval 23068 pi1xfrf 23073 pi1xfrval 23074 pi1xfrcnvlem 23076 pi1xfrcnv 23077 pi1cof 23079 pi1coval 23080 pi1coghm 23081 vitalilem3 23598 ismntoplly 30399 linedegen 32577 fvline 32578 |
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