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

Theorem csbgfi 3832
Description: Substitution for a variable not free in a class does not affect it, in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
Hypotheses
Ref Expression
csbgfi.1 𝐴 ∈ V
csbgfi.2 𝑥𝐵
Assertion
Ref Expression
csbgfi 𝐴 / 𝑥𝐵 = 𝐵

Proof of Theorem csbgfi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3812 . . . 4 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
21abeq2i 2872 . . 3 (𝑦𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝑦𝐵)
3 csbgfi.1 . . . 4 𝐴 ∈ V
4 csbgfi.2 . . . . 5 𝑥𝐵
54nfcri 2891 . . . 4 𝑥 𝑦𝐵
63, 5sbcgfi 3776 . . 3 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵)
72, 6bitri 278 . 2 (𝑦𝐴 / 𝑥𝐵𝑦𝐵)
87eqriv 2734 1 𝐴 / 𝑥𝐵 = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2110  wnfc 2884  Vcvv 3408  [wsbc 3694  csb 3811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-sbc 3695  df-csb 3812
This theorem is referenced by:  fmptdF  30713  sbccom2f  36021
  Copyright terms: Public domain W3C validator