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Theorem cbvcsb 2977
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 2250 . . . 4 𝑦 𝑧𝐶
3 cbvcsb.2 . . . . 5 𝑥𝐷
43nfcri 2250 . . . 4 𝑥 𝑧𝐷
5 cbvcsb.3 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65eleq2d 2185 . . . 4 (𝑥 = 𝑦 → (𝑧𝐶𝑧𝐷))
72, 4, 6cbvsbc 2907 . . 3 ([𝐴 / 𝑥]𝑧𝐶[𝐴 / 𝑦]𝑧𝐷)
87abbii 2231 . 2 {𝑧[𝐴 / 𝑥]𝑧𝐶} = {𝑧[𝐴 / 𝑦]𝑧𝐷}
9 df-csb 2974 . 2 𝐴 / 𝑥𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐶}
10 df-csb 2974 . 2 𝐴 / 𝑦𝐷 = {𝑧[𝐴 / 𝑦]𝑧𝐷}
118, 9, 103eqtr4i 2146 1 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  {cab 2101  wnfc 2243  [wsbc 2880  csb 2973
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-sbc 2881  df-csb 2974
This theorem is referenced by:  cbvcsbv  2978  cbvsum  11080
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