Theorem sbcgfi 3847

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 3844	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
4	1, 3	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  Ⅎwnf 1780   ∈ wcel 2110  Vcvv 3494  [wsbc 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3772

This theorem is referenced by:  csbgfi  3902  bnj110  32125  bnj1039  32238  mptsnunlem  34613  sbali  35384  sbexi  35385