Theorem sbcg 3033

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3031. (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 1528	. 2 |- F/ x ph
2	1	sbcgf 3031	1 |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2148   [.wsbc 2963

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-sbc 2964

This theorem is referenced by:  sbcabel  3045  csbunig  3818  csbxpg  4708  sbcfung  5241  f1od2  6236