Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  qseq1 Structured version   GIF version

Theorem qseq1 6053
 Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
qseq1 (A = B → (A / 𝐶) = (B / 𝐶))

Proof of Theorem qseq1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2475 . . 3 (A = B → (x A y = [x]𝐶x B y = [x]𝐶))
21abbidv 2128 . 2 (A = B → {yx A y = [x]𝐶} = {yx B y = [x]𝐶})
3 df-qs 6011 . 2 (A / 𝐶) = {yx A y = [x]𝐶}
4 df-qs 6011 . 2 (B / 𝐶) = {yx B y = [x]𝐶}
52, 3, 43eqtr4g 2070 1 (A = B → (A / 𝐶) = (B / 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1223  {cab 1999  ∃wrex 2276  [cec 6003   / cqs 6004 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995 This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rex 2281  df-qs 6011 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator