Theorem abbi1dv 1579

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 1286	. 2 |- (ph -> A.x(ps <-> x e. A))
3		abeq1 1569	. 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 954   = wceq 956   e. wcel 958  {cab 1463

This theorem is referenced by:  csbvarg 2021  csbiegft 2029  dffsum 6998  hmeogrp 10538  trran 10636

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472

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