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Theorem sbcgfi 3823
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 3820 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜑))
41, 3ax-mp 5 1 ([𝐴 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wnf 1785  wcel 2115  Vcvv 3471  [wsbc 3749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-sbc 3750
This theorem is referenced by:  csbgfi  3877  bnj110  32138  bnj1039  32251  mptsnunlem  34639  sbali  35432  sbexi  35433
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