Theorem csbeq2 3848

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 2841	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
2	1	alimi 1821	. . . 4 ⊢ (∀𝑥 𝐵 = 𝐶 → ∀𝑥(𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
3		sbcbi2 3793	. . . 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 2818	. 2 ⊢ (∀𝑥 𝐵 = 𝐶 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})
6		df-csb 3844	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
7		df-csb 3844	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2812	1 ⊢ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1548   = wceq 1550   ∈ wcel 2132  {cab 2730  [wsbc 3735  ⦋csb 3843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-sbc 3736  df-csb 3844

This theorem is referenced by:  sumeq2w  15691  prodeq2w  15912  itgeq12i  36504  csbeq12  38595  csbsngVD  45406  csbxpgVD  45407  csbresgVD  45408  csbrngVD  45409  csbima12gALTVD  45410  csbfv12gALTVD  45412