Theorem sbc6 3003

Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Hypothesis

Ref	Expression
sbc6.1	|- A e. _V

Assertion

Ref	Expression
sbc6	|- ( [. A / x ]. ph <-> A. x ( x = A -> ph ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem sbc6

Step	Hyp	Ref	Expression
1		sbc6.1	. 2 |- A e. _V
2		sbc6g 3002	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
3	1, 2	ax-mp 5	1 |- ( [. A / x ]. ph <-> A. x ( x = A -> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2160   _Vcvv 2752   [.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:  intab  3888