MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvcsb Structured version   Visualization version   GIF version

Theorem cbvcsb 3854
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 2370. Use the weaker cbvcsbw 3853 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 2891 . . . 4 𝑦 𝑧𝐶
3 cbvcsb.2 . . . . 5 𝑥𝐷
43nfcri 2891 . . . 4 𝑥 𝑧𝐷
5 cbvcsb.3 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65eleq2d 2822 . . . 4 (𝑥 = 𝑦 → (𝑧𝐶𝑧𝐷))
72, 4, 6cbvsbc 3763 . . 3 ([𝐴 / 𝑥]𝑧𝐶[𝐴 / 𝑦]𝑧𝐷)
87abbii 2806 . 2 {𝑧[𝐴 / 𝑥]𝑧𝐶} = {𝑧[𝐴 / 𝑦]𝑧𝐷}
9 df-csb 3844 . 2 𝐴 / 𝑥𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐶}
10 df-csb 3844 . 2 𝐴 / 𝑦𝐷 = {𝑧[𝐴 / 𝑦]𝑧𝐷}
118, 9, 103eqtr4i 2774 1 𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  {cab 2713  wnfc 2884  [wsbc 3727  csb 3843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-sbc 3728  df-csb 3844
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator