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Theorem qseq2 6444
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 6432 . . . . 5 (𝐴 = 𝐵 → [𝑥]𝐴 = [𝑥]𝐵)
21eqeq2d 2127 . . . 4 (𝐴 = 𝐵 → (𝑦 = [𝑥]𝐴𝑦 = [𝑥]𝐵))
32rexbidv 2413 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐶 𝑦 = [𝑥]𝐴 ↔ ∃𝑥𝐶 𝑦 = [𝑥]𝐵))
43abbidv 2233 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴} = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵})
5 df-qs 6401 . 2 (𝐶 / 𝐴) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴}
6 df-qs 6401 . 2 (𝐶 / 𝐵) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵}
74, 5, 63eqtr4g 2173 1 (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  {cab 2101  wrex 2392  [cec 6393   / cqs 6394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-ec 6397  df-qs 6401
This theorem is referenced by: (None)
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