Theorem abbi1dv 1571

Description: Deduction from a wff to a class abstraction.

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

Proof of Theorem abbi1dv

Step	Hyp	Ref	Expression
1		abbildv.1	. . 3 |- (ph -> (ps <-> x e. A))
2	1	19.21aiv 1281	. 2 |- (ph -> A.x(ps <-> x e. A))
3		abeq1 1561	. 2 |- ({x | ps} = A <-> A.x(ps <-> x e. A))
4	2, 3	sylibr 200	1 |- (ph -> {x | ps} = A)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   = wceq 953   e. wcel 955  {cab 1456

This theorem is referenced by:  csbvarg 2011  csbiegft 2019  dffsum 6936  hmeogrp 10425  trran 10480

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465

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