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Theorem eceq1d 7728
Description: Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
Hypothesis
Ref Expression
eceq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
eceq1d (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)

Proof of Theorem eceq1d
StepHypRef Expression
1 eceq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 eceq1 7727 . 2 (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
31, 2syl 17 1 (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  [cec 7685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ec 7689
This theorem is referenced by:  brecop  7785  eroveu  7787  erov  7789  ecovcom  7799  ecovass  7800  ecovdi  7801  addsrmo  9838  mulsrmo  9839  addsrpr  9840  mulsrpr  9841  supsrlem  9876  supsr  9877  qus0  17573  qusinv  17574  qussub  17575  sylow2blem2  17957  frgpadd  18097  vrgpval  18101  vrgpinv  18103  frgpup3lem  18111  qusabl  18189  quscrng  19159  qustgplem  21834  pi1addval  22756  pi1xfrf  22761  pi1xfrval  22762  pi1xfrcnvlem  22764  pi1xfrcnv  22765  pi1cof  22767  pi1coval  22768  pi1coghm  22769  vitalilem3  23285  ismntoplly  29851  linedegen  31892  fvline  31893
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