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

Theorem csbsng 4241
Description: Distribute proper substitution through the singleton of a class. csbsng 4241 is derived from the virtual deduction proof csbsngVD 38955. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbsng (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})

Proof of Theorem csbsng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbab 4006 . . 3 𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦[𝐴 / 𝑥]𝑦 = 𝐵}
2 sbceq2g 3988 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵))
32abbidv 2740 . . 3 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
41, 3syl5eq 2667 . 2 (𝐴𝑉𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
5 df-sn 4176 . . 3 {𝐵} = {𝑦𝑦 = 𝐵}
65csbeq2i 3991 . 2 𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}
7 df-sn 4176 . 2 {𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}
84, 6, 73eqtr4g 2680 1 (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  {cab 2607  [wsbc 3433  csb 3531  {csn 4175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-nul 3914  df-sn 4176
This theorem is referenced by:  csbprg  4242  csbopg  4418  csbpredg  33152  csbfv12gALTOLD  38878  csbfv12gALTVD  38961
  Copyright terms: Public domain W3C validator