Theorem rabid2f 3119

Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)

Hypothesis

Ref	Expression
rabid2f.1	|- F/_ x A

Assertion

Ref	Expression
rabid2f	|- ( A = { x e. A | ph } <-> A. x e. A ph )

Proof of Theorem rabid2f

Step	Hyp	Ref	Expression
1		rabid2f.1	. . . 4 |- F/_ x A
2	1	abeq2f 2792	. . 3 |- ( A = { x | ( x e. A /\ ph ) } <-> A. x ( x e. A <-> ( x e. A /\ ph ) ) )
3		pm4.71 662	. . . 4 |- ( ( x e. A -> ph ) <-> ( x e. A <-> ( x e. A /\ ph ) ) )
4	3	albii 1747	. . 3 |- ( A. x ( x e. A -> ph ) <-> A. x ( x e. A <-> ( x e. A /\ ph ) ) )
5	2, 4	bitr4i 267	. 2 |- ( A = { x | ( x e. A /\ ph ) } <-> A. x ( x e. A -> ph ) )
6		df-rab 2921	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	6	eqeq2i 2634	. 2 |- ( A = { x e. A | ph } <-> A = { x | ( x e. A /\ ph ) } )
8		df-ral 2917	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
9	5, 7, 8	3bitr4i 292	1 |- ( A = { x e. A | ph } <-> A. x e. A ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921

This theorem is referenced by:  funcnvmptOLD  29467  funcnvmpt  29468  dmmptdf2  39439