Theorem sbcg 3059

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3057. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
sbcg	|- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   V( x)

Proof of Theorem sbcg

Step	Hyp	Ref	Expression
1		nfv 1542	. 2 |- F/ x ph
2	1	sbcgf 3057	1 |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2167   [.wsbc 2989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  sbcabel  3071  csbunig  3847  csbxpg  4744  sbcfung  5282  f1od2  6293