Theorem abbi1dv 2286

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 1862	. 2 |- ( ph -> A. x ( ps <-> x e. A ) )
3		abeq1 2276	. 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 1341   = wceq 1343   e. wcel 2136   {cab 2151

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161

This theorem is referenced by:  abidnf  2894  csbtt  3057  csbvarg  3073  csbie2g  3095  abvor0dc  3432  iinxsng  3939  shftuz  10759