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Theorem sbcgf 3719
 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 3718 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))
31, 2mpan2 681 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  Ⅎwnf 1827   ∈ wcel 2107  [wsbc 3652 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-v 3400  df-sbc 3653 This theorem is referenced by:  sbc19.21g  3720  sbcg  3721  sbcabel  3734  bnj110  31531  bnj1039  31642  sbali  34543  sbexi  34544  sbcgfi  34560
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