Theorem eceq2 8363

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 5899	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2		df-ec 8325	. 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶})
3		df-ec 8325	. 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶})
4	1, 2, 3	3eqtr4g 2799	1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)

Syntax hints:   → wi 4   = wceq 1542  {csn 4517   “ cima 5529  [cec 8321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3401  df-un 3849  df-in 3851  df-ss 3861  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-cnv 5534  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ec 8325

This theorem is referenced by:  eceq2i  8364  eceq2d  8365  qseq2  8378  qusval  16921  efgrelexlemb  18997  efgcpbllemb  19002  znzrh2  20367