Theorem sbcg 3020

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3018. (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 1516	. 2 |- F/ x ph
2	1	sbcgf 3018	1 |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2136   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by:  sbcabel  3032  csbunig  3797  csbxpg  4685  sbcfung  5212  f1od2  6203