Theorem abbi1dv 2157

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 1754	. 2 |- ( ph -> A. x ( ps <-> x e. A ) )
3		abeq1 2147	. 2 |- ( { x | ps } = A <-> A. x ( ps <-> x e. A ) )
4	2, 3	sylibr 137	1 |- ( ph -> { x | ps } = A )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036

This theorem is referenced by:  abidnf  2709  csbtt  2862  csbvarg  2877  csbie2g  2896  abvor0dc  3242  iinxsng  3730  shftuz  9418