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Theorem cbvcsb 3062
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 2313 . . . 4 𝑦 𝑧𝐶
3 cbvcsb.2 . . . . 5 𝑥𝐷
43nfcri 2313 . . . 4 𝑥 𝑧𝐷
5 cbvcsb.3 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65eleq2d 2247 . . . 4 (𝑥 = 𝑦 → (𝑧𝐶𝑧𝐷))
72, 4, 6cbvsbc 2991 . . 3 ([𝐴 / 𝑥]𝑧𝐶[𝐴 / 𝑦]𝑧𝐷)
87abbii 2293 . 2 {𝑧[𝐴 / 𝑥]𝑧𝐶} = {𝑧[𝐴 / 𝑦]𝑧𝐷}
9 df-csb 3058 . 2 𝐴 / 𝑥𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐶}
10 df-csb 3058 . 2 𝐴 / 𝑦𝐷 = {𝑧[𝐴 / 𝑦]𝑧𝐷}
118, 9, 103eqtr4i 2208 1 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  {cab 2163  wnfc 2306  [wsbc 2962  csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-sbc 2963  df-csb 3058
This theorem is referenced by:  cbvcsbv  3063  cbvsum  11367
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