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Theorem cbvcsb 3140
 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 / x]C = [A / y]D

Proof of Theorem cbvcsb
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 cbvcsb.1 . . . . 5 yC
21nfcri 2483 . . . 4 y z C
3 cbvcsb.2 . . . . 5 xD
43nfcri 2483 . . . 4 x z D
5 cbvcsb.3 . . . . 5 (x = yC = D)
65eleq2d 2420 . . . 4 (x = y → (z Cz D))
72, 4, 6cbvsbc 3074 . . 3 ([̣A / xz C ↔ [̣A / yz D)
87abbii 2465 . 2 {z A / xz C} = {z A / yz D}
9 df-csb 3137 . 2 [A / x]C = {z A / xz C}
10 df-csb 3137 . 2 [A / y]D = {z A / yz D}
118, 9, 103eqtr4i 2383 1 [A / x]C = [A / y]D
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-sbc 3047  df-csb 3137 This theorem is referenced by:  cbvcsbv  3141
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