Theorem sbcgfi 3857

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 3853	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
4	1, 3	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 205  Ⅎwnf 1778   ∈ wcel 2099  Vcvv 3471  [wsbc 3776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-sbc 3777

This theorem is referenced by:  csbgfi  3913  bnj110  34489  bnj1039  34602  mptsnunlem  36817  sbali  37585  sbexi  37586

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