Theorem abbi1dv 2351

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 1922	. 2 |- ( ph -> A. x ( ps <-> x e. A ) )
3		abeq1 2341	. 2 |- ( { x | ps } = A <-> A. x ( ps <-> x e. A ) )
4	2, 3	sylibr 134	1 |- ( ph -> { x | ps } = A )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1395   = wceq 1397   e. wcel 2202   {cab 2217

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-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227

This theorem is referenced by:  abidnf  2974  csbtt  3139  csbvarg  3155  csbie2g  3178  abvor0dc  3518  iinxsng  4044  shftuz  11377