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

Theorem sbcgfi 33103
 Description: Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
Hypotheses
Ref Expression
sbcgfi.1 𝐴 ∈ V
sbcgfi.2 𝑥𝜑
Assertion
Ref Expression
sbcgfi ([𝐴 / 𝑥]𝜑𝜑)

Proof of Theorem sbcgfi
StepHypRef Expression
1 sbcgfi.1 . 2 𝐴 ∈ V
2 sbcgfi.2 . . 3 𝑥𝜑
32sbcgf 3468 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜑))
41, 3ax-mp 5 1 ([𝐴 / 𝑥]𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  Ⅎwnf 1699   ∈ wcel 1977  Vcvv 3173  [wsbc 3402 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-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-v 3175  df-sbc 3403 This theorem is referenced by:  csbgfi  33105
 Copyright terms: Public domain W3C validator