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Theorem sbcgfi 3872
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 3868 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜑))
41, 3ax-mp 5 1 ([𝐴 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wnf 1780  wcel 2106  Vcvv 3478  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792
This theorem is referenced by:  csbgfi  3929  bnj110  34851  bnj1039  34964  mptsnunlem  37321  sbali  38099  sbexi  38100
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