Theorem rabid2 2667

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 2298	. . 3 |- ( A = { x | ( x e. A /\ ph ) } <-> A. x ( x e. A <-> ( x e. A /\ ph ) ) )
2		pm4.71 389	. . . 4 |- ( ( x e. A -> ph ) <-> ( x e. A <-> ( x e. A /\ ph ) ) )
3	2	albii 1481	. . 3 |- ( A. x ( x e. A -> ph ) <-> A. x ( x e. A <-> ( x e. A /\ ph ) ) )
4	1, 3	bitr4i 187	. 2 |- ( A = { x | ( x e. A /\ ph ) } <-> A. x ( x e. A -> ph ) )
5		df-rab 2477	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eqeq2i 2200	. 2 |- ( A = { x e. A | ph } <-> A = { x | ( x e. A /\ ph ) } )
7		df-ral 2473	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
8	4, 6, 7	3bitr4i 212	1 |- ( A = { x e. A | ph } <-> A. x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2160   {cab 2175   A.wral 2468   {crab 2472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-rab 2477

This theorem is referenced by:  rabxmdc  3469  rabrsndc  3675  class2seteq  4178  dmmptg  5141  dmmptd  5361  fneqeql  5640  fmpt  5682  acexmidlemph  5884  inffiexmid  6924  ssfirab  6951  exmidaclem  7225  ioomax  9966  iccmax  9967  dfphi2  12238  phiprmpw  12240  phisum  12258  unennn  12416  znnen  12417  isnsg4  13117  lssuni  13640