Theorem qseq2 8677

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.

Step	Hyp	Ref	Expression
1		eceq2 8658	. . . . 5 ⊢ (𝐴 = 𝐵 → [𝑥]𝐴 = [𝑥]𝐵)
2	1	eqeq2d 2742	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑦 = [𝑥]𝐴 ↔ 𝑦 = [𝑥]𝐵))
3	2	rexbidv 3156	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐴 ↔ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐵))
4	3	abbidv 2797	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐵})
5		df-qs 8623	. 2 ⊢ (𝐶 / 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐴}
6		df-qs 8623	. 2 ⊢ (𝐶 / 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐵}
7	4, 5, 6	3eqtr4g 2791	1 ⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))

Syntax hints:   → wi 4   = wceq 1541  {cab 2709  ∃wrex 3056  [cec 8615   / cqs 8616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ec 8619  df-qs 8623

This theorem is referenced by:  qseq2i  8678  qseq2d  8680  qseq12  8681  pi1bas3  24965