Theorem sbc6g 3022

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 3021	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
2		nfe1 1518	. . 3 |- F/ x E. x ( x = A /\ ph )
3		ceqex 2899	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
4	2, 3	ceqsalg 2799	. 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 1370   = wceq 1372   E.wex 1514   e. wcel 2175   [.wsbc 2997

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998

This theorem is referenced by:  sbc6  3023  sbciegft  3028  ralsnsg  3669  ralsns  3670  fz1sbc  10217