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Theorem qseq2 5975
 Description: Equality theorem for quotient set. (Contributed by set.mm contributors, 23-Jul-1995.)
Assertion
Ref Expression
qseq2 (A = B → (C / A) = (C / B))

Proof of Theorem qseq2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eceq2 5963 . . . . 5 (A = B → [x]A = [x]B)
21eqeq2d 2364 . . . 4 (A = B → (y = [x]Ay = [x]B))
32rexbidv 2635 . . 3 (A = B → (x C y = [x]Ax C y = [x]B))
43abbidv 2467 . 2 (A = B → {y x C y = [x]A} = {y x C y = [x]B})
5 df-qs 5951 . 2 (C / A) = {y x C y = [x]A}
6 df-qs 5951 . 2 (C / B) = {y x C y = [x]B}
74, 5, 63eqtr4g 2410 1 (A = B → (C / A) = (C / B))
 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-1 5  ax-2 6  ax-3 7  ax-mp 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2620  df-br 4640  df-ima 4727  df-ec 5947  df-qs 5951 This theorem is referenced by: (None)
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