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

Theorem sbcgfi 32899
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 3467 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜑))
41, 3ax-mp 5 1 ([𝐴 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wnf 1698  wcel 1976  Vcvv 3172  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-v 3174  df-sbc 3402
This theorem is referenced by:  csbgfi  32901
  Copyright terms: Public domain W3C validator