Theorem sbcgf 3850

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	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
sbcgf	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Proof of Theorem sbcgf

Step	Hyp	Ref	Expression
1		sbcgf.1	. 2 ⊢ Ⅎ𝑥𝜑
2		sbctt 3849	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	mpan2 690	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1778   ∈ wcel 2099  [wsbc 3774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-sbc 3775

This theorem is referenced by:  sbc19.21g  3851  sbcgOLD  3853  sbcgfi  3854  sbcabel  3868  2nreu  4437