Theorem sbc6g 2853

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		nfe1 1428	. . 3 |- F/ x E. x ( x = A /\ ph )
2		ceqex 2735	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
3	1, 2	ceqsalg 2641	. 2 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
4		sbc5 2852	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
5	3, 4	syl6rbbr 197	1 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1285   = wceq 1287   E.wex 1424   e. wcel 1436   [.wsbc 2829

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830

This theorem is referenced by:  sbc6  2854  sbciegft  2858  ralsnsg  3465  ralsns  3466  fz1sbc  9443