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Theorem csbeq2 3055
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 2221 . . . . 5 (𝐵 = 𝐶 → (𝑦𝐵𝑦𝐶))
21alimi 1435 . . . 4 (∀𝑥 𝐵 = 𝐶 → ∀𝑥(𝑦𝐵𝑦𝐶))
3 sbcbi2 2987 . . . 4 (∀𝑥(𝑦𝐵𝑦𝐶) → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
42, 3syl 14 . . 3 (∀𝑥 𝐵 = 𝐶 → ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2275 . 2 (∀𝑥 𝐵 = 𝐶 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
6 df-csb 3032 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
7 df-csb 3032 . 2 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
85, 6, 73eqtr4g 2215 1 (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1333   = wceq 1335  wcel 2128  {cab 2143  [wsbc 2937  csb 3031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-sbc 2938  df-csb 3032
This theorem is referenced by:  prodeq2w  11457
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