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Theorem eceq1d 7952
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 7951 . 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 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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