Theorem sbcgfi 3859

Description: Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)

Hypotheses

Ref	Expression
sbcgfi.1	⊢ 𝐴 ∈ V
sbcgfi.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
sbcgfi	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

Proof of Theorem sbcgfi

Step	Hyp	Ref	Expression
1		sbcgfi.1	. 2 ⊢ 𝐴 ∈ V
2		sbcgfi.2	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	sbcgf 3855	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
4	1, 3	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  Ⅎwnf 1786   ∈ wcel 2107  Vcvv 3475  [wsbc 3778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sbc 3779

This theorem is referenced by:  csbgfi  3915  bnj110  33869  bnj1039  33982  mptsnunlem  36219  sbali  36980  sbexi  36981