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Theorem cbvcsb 2850
Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
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 2169 . . . 4 y z 𝐶
3 cbvcsb.2 . . . . 5 x𝐷
43nfcri 2169 . . . 4 x z 𝐷
5 cbvcsb.3 . . . . 5 (x = y𝐶 = 𝐷)
65eleq2d 2104 . . . 4 (x = y → (z 𝐶z 𝐷))
72, 4, 6cbvsbc 2785 . . 3 ([A / x]z 𝐶[A / y]z 𝐷)
87abbii 2150 . 2 {z[A / x]z 𝐶} = {z[A / y]z 𝐷}
9 df-csb 2847 . 2 A / x𝐶 = {z[A / x]z 𝐶}
10 df-csb 2847 . 2 A / y𝐷 = {z[A / y]z 𝐷}
118, 9, 103eqtr4i 2067 1 A / x𝐶 = A / y𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  {cab 2023  wnfc 2162  [wsbc 2758  csb 2846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-sbc 2759  df-csb 2847
This theorem is referenced by:  cbvcsbv  2851
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