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Theorem sbcgf 3651
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 3650 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))
31, 2mpan2 671 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1856  wcel 2145  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by:  sbc19.21g  3652  sbcg  3653  sbcabel  3666  bnj110  31266  bnj1039  31377  sbali  34247  sbexi  34248  sbcgfi  34265
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