Theorem sbc6g 2975

Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
sbc6g	|- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )

Distinct variable group:   x, A

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

Proof of Theorem sbc6g

Step	Hyp	Ref	Expression
1		sbc5 2974	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
2		nfe1 1484	. . 3 |- F/ x E. x ( x = A /\ ph )
3		ceqex 2853	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
4	2, 3	ceqsalg 2754	. 2 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
5	1, 4	bitr4id 198	1 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   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:  sbc6  2976  sbciegft  2981  ralsnsg  3613  ralsns  3614  fz1sbc  10031