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Theorem qseq2 8801
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 8785 . . . . 5 (𝐴 = 𝐵 → [𝑥]𝐴 = [𝑥]𝐵)
21eqeq2d 2746 . . . 4 (𝐴 = 𝐵 → (𝑦 = [𝑥]𝐴𝑦 = [𝑥]𝐵))
32rexbidv 3177 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐶 𝑦 = [𝑥]𝐴 ↔ ∃𝑥𝐶 𝑦 = [𝑥]𝐵))
43abbidv 2806 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴} = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵})
5 df-qs 8750 . 2 (𝐶 / 𝐴) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴}
6 df-qs 8750 . 2 (𝐶 / 𝐵) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵}
74, 5, 63eqtr4g 2800 1 (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  {cab 2712  wrex 3068  [cec 8742   / cqs 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750
This theorem is referenced by:  qseq2i  8802  qseq2d  8804  qseq12  8805  pi1bas3  25090
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