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Theorem qseq2 8704
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 8689 . . . . 5 (𝐴 = 𝐵 → [𝑥]𝐴 = [𝑥]𝐵)
21eqeq2d 2748 . . . 4 (𝐴 = 𝐵 → (𝑦 = [𝑥]𝐴𝑦 = [𝑥]𝐵))
32rexbidv 3176 . . 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 8655 . 2 (𝐶 / 𝐴) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐴}
6 df-qs 8655 . 2 (𝐶 / 𝐵) = {𝑦 ∣ ∃𝑥𝐶 𝑦 = [𝑥]𝐵}
74, 5, 63eqtr4g 2802 1 (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  {cab 2714  wrex 3074  [cec 8647   / cqs 8648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651  df-qs 8655
This theorem is referenced by:  qseq2i  8705  qseq2d  8706  qseq12  8707  pi1bas3  24409
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