Theorem sbc6g 2900

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 1453	. . 3 |- F/ x E. x ( x = A /\ ph )
2		ceqex 2780	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
3	1, 2	ceqsalg 2683	. 2 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
4		sbc5 2899	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
5	3, 4	syl6rbbr 198	1 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1310   = wceq 1312   E.wex 1449   e. wcel 1461   [.wsbc 2876

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-sbc 2877

This theorem is referenced by:  sbc6  2901  sbciegft  2905  ralsnsg  3525  ralsns  3526  fz1sbc  9763