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Theorem cbvcsb 3816
 Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvcsbw 3815 when possible. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvcsb.1 𝑦𝐶
cbvcsb.2 𝑥𝐷
cbvcsb.3 (𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
cbvcsb 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷

Proof of Theorem cbvcsb
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvcsb.1 . . . . 5 𝑦𝐶
21nfcri 2906 . . . 4 𝑦 𝑧𝐶
3 cbvcsb.2 . . . . 5 𝑥𝐷
43nfcri 2906 . . . 4 𝑥 𝑧𝐷
5 cbvcsb.3 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65eleq2d 2837 . . . 4 (𝑥 = 𝑦 → (𝑧𝐶𝑧𝐷))
72, 4, 6cbvsbc 3731 . . 3 ([𝐴 / 𝑥]𝑧𝐶[𝐴 / 𝑦]𝑧𝐷)
87abbii 2823 . 2 {𝑧[𝐴 / 𝑥]𝑧𝐶} = {𝑧[𝐴 / 𝑦]𝑧𝐷}
9 df-csb 3806 . 2 𝐴 / 𝑥𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐶}
10 df-csb 3806 . 2 𝐴 / 𝑦𝐷 = {𝑧[𝐴 / 𝑦]𝑧𝐷}
118, 9, 103eqtr4i 2791 1 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {cab 2735  Ⅎwnfc 2899  [wsbc 3696  ⦋csb 3805 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-sbc 3697  df-csb 3806 This theorem is referenced by: (None)
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