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Theorem qseq2 6578
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 6566 . . . . 5 (𝐴 = 𝐵 → [𝑥]𝐴 = [𝑥]𝐵)
21eqeq2d 2189 . . . 4 (𝐴 = 𝐵 → (𝑦 = [𝑥]𝐴𝑦 = [𝑥]𝐵))
32rexbidv 2478 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐶 𝑦 = [𝑥]𝐴 ↔ ∃𝑥𝐶 𝑦 = [𝑥]𝐵))
43abbidv 2295 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴} = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵})
5 df-qs 6535 . 2 (𝐶 / 𝐴) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴}
6 df-qs 6535 . 2 (𝐶 / 𝐵) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵}
74, 5, 63eqtr4g 2235 1 (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  {cab 2163  wrex 2456  [cec 6527   / cqs 6528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-cnv 4631  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-ec 6531  df-qs 6535
This theorem is referenced by: (None)
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