Theorem sbc6g 3002

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 3001	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
2		nfe1 1507	. . 3 |- F/ x E. x ( x = A /\ ph )
3		ceqex 2879	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
4	2, 3	ceqsalg 2780	. 2 |- ( A e. V -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
5	1, 4	bitr4id 199	1 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2160   [.wsbc 2977

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978

This theorem is referenced by:  sbc6  3003  sbciegft  3008  ralsnsg  3644  ralsns  3645  fz1sbc  10128