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Theorem csbeq2 3502
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 2676 . . . . 5 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
21alimi 1729 . . . 4 (∀𝑥 𝐵 = 𝐶 → ∀𝑥(𝑦𝐵𝑦𝐶))
3 sbcbi2 3450 . . . 4 (∀𝑥(𝑦𝐵𝑦𝐶) → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
42, 3syl 17 . . 3 (∀𝑥 𝐵 = 𝐶 → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2727 . 2 (∀𝑥 𝐵 = 𝐶 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
6 df-csb 3499 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
7 df-csb 3499 . 2 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
85, 6, 73eqtr4g 2668 1 (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472   = wceq 1474  wcel 1976  {cab 2595  [wsbc 3401  csb 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-sbc 3402  df-csb 3499
This theorem is referenced by:  sumeq2w  14216  prodeq2w  14427  csbeq12  32939  csbfv12gALTVD  37960
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