Theorem abbi1dv 2290

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 1867	. 2 |- ( ph -> A. x ( ps <-> x e. A ) )
3		abeq1 2280	. 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 1346   = wceq 1348   e. wcel 2141   {cab 2156

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166

This theorem is referenced by:  abidnf  2898  csbtt  3061  csbvarg  3077  csbie2g  3099  abvor0dc  3438  iinxsng  3946  shftuz  10781