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Theorem eceq2 7951
 Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
eceq2 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)

Proof of Theorem eceq2
StepHypRef Expression
1 imaeq1 5619 . 2 (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2 df-ec 7913 . 2 [𝐶]𝐴 = (𝐴 “ {𝐶})
3 df-ec 7913 . 2 [𝐶]𝐵 = (𝐵 “ {𝐶})
41, 2, 33eqtr4g 2819 1 (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632  {csn 4321   “ cima 5269  [cec 7909 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-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ec 7913 This theorem is referenced by:  qseq2  7964  qusval  16404  efgrelexlemb  18363  efgcpbllemb  18368  vrgpfval  18379  znzrh2  20096  eceq2i  34364  eceq2d  34365
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