Theorem rabid2 2607

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

Assertion

Ref	Expression
rabid2	|- ( A = { x e. A | ph } <-> A. x e. A ph )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		abeq2 2248	. . 3 |- ( A = { x | ( x e. A /\ ph ) } <-> A. x ( x e. A <-> ( x e. A /\ ph ) ) )
2		pm4.71 386	. . . 4 |- ( ( x e. A -> ph ) <-> ( x e. A <-> ( x e. A /\ ph ) ) )
3	2	albii 1446	. . 3 |- ( A. x ( x e. A -> ph ) <-> A. x ( x e. A <-> ( x e. A /\ ph ) ) )
4	1, 3	bitr4i 186	. 2 |- ( A = { x | ( x e. A /\ ph ) } <-> A. x ( x e. A -> ph ) )
5		df-rab 2425	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eqeq2i 2150	. 2 |- ( A = { x e. A | ph } <-> A = { x | ( x e. A /\ ph ) } )
7		df-ral 2421	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
8	4, 6, 7	3bitr4i 211	1 |- ( A = { x e. A | ph } <-> A. x e. A ph )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   {crab 2420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-rab 2425

This theorem is referenced by:  rabxmdc  3394  rabrsndc  3591  class2seteq  4087  dmmptg  5036  dmmptd  5253  fneqeql  5528  fmpt  5570  acexmidlemph  5767  inffiexmid  6800  ssfirab  6822  exmidaclem  7064  ioomax  9731  iccmax  9732  dfphi2  11896  phiprmpw  11898  unennn  11910  znnen  11911