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Theorem qseq1 5974
 Description: Equality theorem for quotient set. (Contributed by set.mm contributors, 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 2808 . . 3 (A = B → (x A y = [x]Cx B y = [x]C))
21abbidv 2467 . 2 (A = B → {y x A y = [x]C} = {y x B y = [x]C})
3 df-qs 5951 . 2 (A / C) = {y x A y = [x]C}
4 df-qs 5951 . 2 (B / C) = {y x B y = [x]C}
52, 3, 43eqtr4g 2410 1 (A = B → (A / C) = (B / C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  {cab 2339  ∃wrex 2615  [cec 5945   / cqs 5946 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-qs 5951 This theorem is referenced by: (None)
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