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Theorem csbeq2 3503
 Description: Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
Assertion
Ref Expression
csbeq2 (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)

Proof of Theorem csbeq2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2677 . . . . 5 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
21alimi 1730 . . . 4 (∀𝑥 𝐵 = 𝐶 → ∀𝑥(𝑦𝐵𝑦𝐶))
3 sbcbi2 3451 . . . 4 (∀𝑥(𝑦𝐵𝑦𝐶) → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
42, 3syl 17 . . 3 (∀𝑥 𝐵 = 𝐶 → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2728 . 2 (∀𝑥 𝐵 = 𝐶 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
6 df-csb 3500 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
7 df-csb 3500 . 2 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
85, 6, 73eqtr4g 2669 1 (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  [wsbc 3402  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403  df-csb 3500 This theorem is referenced by:  sumeq2w  14270  prodeq2w  14481  csbeq12  33136  csbfv12gALTVD  38157
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