Theorem sbcgfi 3811

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 3808	. 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 1815   ∈ wcel 2144  Vcvv 3450  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-sbc 3739

This theorem is used by:  csbgfi  3866  bnj110  35488  bnj1039  35601  mptsnunlem  38262  sbali  39045  sbexi  39046