Theorem sbcgf 3814

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 3813	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	mpan2 703	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1813   ∈ wcel 2143  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-sbc 3745

This theorem is referenced by:  sbc19.21g  3815  sbcgfi  3817  sbcabel  3831  2nreu  4409  modelaxreplem3  45689