Theorem sbcgf 3789

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 3788	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	mpan2 687	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1787   ∈ wcel 2108  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-sbc 3712

This theorem is referenced by:  sbc19.21g  3790  sbcgOLD  3792  sbcgfi  3793  sbcabel  3807  2nreu  4372