Theorem eceq2 6626

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 5001	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶}))
2		df-ec 6591	. 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶})
3		df-ec 6591	. 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶})
4	1, 2, 3	3eqtr4g 2251	1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)

Syntax hints:   → wi 4   = wceq 1364  {csn 3619   “ cima 4663  [cec 6587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-ec 6591

This theorem is referenced by:  eceq2i  6627  eceq2d  6628  qseq2  6640  nqnq0pi  7500  qusval  12909  qusex  12911  znzrh2  14145