Theorem abbi1dv 2154

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 1751	. 2 |- (ph -> A.x(ps <-> x e. A))
3		abeq1 2144	. 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 1240   = wceq 1242   e. wcel 1390   {cab 2023

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033

This theorem is referenced by:  abidnf  2703  csbtt  2856  csbvarg  2871  csbie2g  2890  abvor0dc  3236  iinxsng  3721