Theorem sbc6g 3053

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 3052	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
2		nfe1 1542	. . 3 |- F/ x E. x ( x = A /\ ph )
3		ceqex 2930	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
4	2, 3	ceqsalg 2828	. 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 1393   = wceq 1395   E.wex 1538   e. wcel 2200   [.wsbc 3028

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029

This theorem is referenced by:  sbc6  3054  sbciegft  3059  ralsnsg  3703  ralsns  3704  fz1sbc  10288