Theorem sbcgfi 3815

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 3812	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
4	1, 3	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  Ⅎwnf 1816   ∈ wcel 2145  Vcvv 3453  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3743

This theorem is used by:  csbgfi  3870  bnj110  35354  bnj1039  35467  mptsnunlem  38079  sbali  38847  sbexi  38848