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Theorem qseq2 6186
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 6174 . . . . 5 (𝐴 = 𝐵 → [𝑥]𝐴 = [𝑥]𝐵)
21eqeq2d 2067 . . . 4 (𝐴 = 𝐵 → (𝑦 = [𝑥]𝐴𝑦 = [𝑥]𝐵))
32rexbidv 2344 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐶 𝑦 = [𝑥]𝐴 ↔ ∃𝑥𝐶 𝑦 = [𝑥]𝐵))
43abbidv 2171 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴} = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵})
5 df-qs 6143 . 2 (𝐶 / 𝐴) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴}
6 df-qs 6143 . 2 (𝐶 / 𝐵) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵}
74, 5, 63eqtr4g 2113 1 (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  {cab 2042  wrex 2324  [cec 6135   / cqs 6136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-ec 6139  df-qs 6143
This theorem is referenced by: (None)
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