Theorem csbeq2 3898

Description: Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Assertion

Ref	Expression
csbeq2	⊢ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem csbeq2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2822	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
2	1	alimi 1813	. . . 4 ⊢ (∀𝑥 𝐵 = 𝐶 → ∀𝑥(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
3		sbcbi2 3839	. . . 4 ⊢ (∀𝑥(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
4	2, 3	syl 17	. . 3 ⊢ (∀𝑥 𝐵 = 𝐶 → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
5	4	abbidv 2801	. 2 ⊢ (∀𝑥 𝐵 = 𝐶 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6		df-csb 3894	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
7		df-csb 3894	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2797	1 ⊢ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2709  [wsbc 3777  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-sbc 3778  df-csb 3894

This theorem is referenced by:  sumeq2w  15642  prodeq2w  15860  csbeq12  37329  csbsngVD  43956  csbxpgVD  43957  csbresgVD  43958  csbrngVD  43959  csbima12gALTVD  43960  csbfv12gALTVD  43962