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Theorem qseq2 6054
Description: Equality theorem for quotient set. (Contributed by NM, 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 6042 . . . . 5 (A = B → [x]A = [x]B)
21eqeq2d 2024 . . . 4 (A = B → (y = [x]Ay = [x]B))
32rexbidv 2296 . . 3 (A = B → (x 𝐶 y = [x]Ax 𝐶 y = [x]B))
43abbidv 2128 . 2 (A = B → {yx 𝐶 y = [x]A} = {yx 𝐶 y = [x]B})
5 df-qs 6011 . 2 (𝐶 / A) = {yx 𝐶 y = [x]A}
6 df-qs 6011 . 2 (𝐶 / B) = {yx 𝐶 y = [x]B}
74, 5, 63eqtr4g 2070 1 (A = B → (𝐶 / A) = (𝐶 / B))
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-3an 869  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-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-cnv 4268  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-ec 6007  df-qs 6011
This theorem is referenced by: (None)
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