Theorem sbcg 3067

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3065. (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 1550	. 2 |- F/ x ph
2	1	sbcgf 3065	1 |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2175   [.wsbc 2997

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998

This theorem is referenced by:  sbcabel  3079  csbunig  3857  csbxpg  4755  sbcfung  5294  f1od2  6320