Theorem abbi1dv 2260

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Hypothesis

Ref	Expression
abbildv.1	|- ( ph -> ( ps <-> x e. A ) )

Assertion

Ref	Expression
abbi1dv	|- ( ph -> { x | ps } = A )

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem abbi1dv

Step	Hyp	Ref	Expression
1		abbildv.1	. . 3 |- ( ph -> ( ps <-> x e. A ) )
2	1	alrimiv 1847	. 2 |- ( ph -> A. x ( ps <-> x e. A ) )
3		abeq1 2250	. 2 |- ( { x | ps } = A <-> A. x ( ps <-> x e. A ) )
4	2, 3	sylibr 133	1 |- ( ph -> { x | ps } = A )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   = wceq 1332   e. wcel 1481   {cab 2126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136

This theorem is referenced by:  abidnf  2856  csbtt  3019  csbvarg  3035  csbie2g  3055  abvor0dc  3391  iinxsng  3894  shftuz  10621