Mathbox for Giovanni Mascellani < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  csbgfi Structured version   Visualization version   GIF version

Theorem csbgfi 33105
 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 3500 . . . 4 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
21abeq2i 2722 . . 3 (𝑦𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝑦𝐵)
3 csbgfi.1 . . . 4 𝐴 ∈ V
4 csbgfi.2 . . . . 5 𝑥𝐵
54nfcri 2745 . . . 4 𝑥 𝑦𝐵
63, 5sbcgfi 33103 . . 3 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵)
72, 6bitri 263 . 2 (𝑦𝐴 / 𝑥𝐵𝑦𝐵)
87eqriv 2607 1 𝐴 / 𝑥𝐵 = 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173  [wsbc 3402  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator