Theorem eceq2 8679

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

Step	Hyp	Ref	Expression
1		imaeq1 6014	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2		df-ec 8639	. 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶})
3		df-ec 8639	. 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶})
4	1, 2, 3	3eqtr4g 2801	1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)

Syntax hints:   → wi 4   = wceq 1548  {csn 4558   “ cima 5624  [cec 8635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8639

This theorem is referenced by:  eceq2i  8680  eceq2d  8681  qseq2  8698  qusval  17501  efgrelexlemb  19720  efgcpbllemb  19725  znzrh2  21524