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Theorem sbcgf 3847
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 3846 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))
31, 2mpan2 688 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wnf 1777  wcel 2098  [wsbc 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-sbc 3771
This theorem is referenced by:  sbc19.21g  3848  sbcgOLD  3850  sbcgfi  3851  sbcabel  3865  2nreu  4434
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