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Theorem sbcgf 3861
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypothesis
Ref Expression
sbcgf.1 𝑥𝜑
Assertion
Ref Expression
sbcgf (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))

Proof of Theorem sbcgf
StepHypRef Expression
1 sbcgf.1 . 2 𝑥𝜑
2 sbctt 3860 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))
31, 2mpan2 691 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wnf 1783  wcel 2108  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789
This theorem is referenced by:  sbc19.21g  3862  sbcgfi  3864  sbcabel  3878  2nreu  4444  modelaxreplem3  44997
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