Theorem sbcgfi 33933

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 e. _V
sbcgfi.2	|- F/ x ph

Assertion

Ref	Expression
sbcgfi	|- ( [. A / x ]. ph <-> ph )

Proof of Theorem sbcgfi

Step	Hyp	Ref	Expression
1		sbcgfi.1	. 2 |- A e. _V
2		sbcgfi.2	. . 3 |- F/ x ph
3	2	sbcgf 3501	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> ph ) )
4	1, 3	ax-mp 5	1 |- ( [. A / x ]. ph <-> ph )

Syntax hints:   <-> wb 196   F/wnf 1708   e. wcel 1990   _Vcvv 3200   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  csbgfi  33935