Theorem rabid2 2729

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 2347	. . 3 |- ( A = { x | ( x e. A /\ ph ) } <-> A. x ( x e. A <-> ( x e. A /\ ph ) ) )
2		pm4.71 393	. . . 4 |- ( ( x e. A -> ph ) <-> ( x e. A <-> ( x e. A /\ ph ) ) )
3	2	albii 1523	. . 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 2537	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eqeq2i 2249	. 2 |- ( A = { x e. A | ph } <-> A = { x | ( x e. A /\ ph ) } )
7		df-ral 2533	. 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 1400   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   {crab 2532

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-rab 2537

This theorem is referenced by:  rabxmdc  3554  rabrsndc  3775  class2seteq  4295  dmmptg  5280  dmmptd  5509  fneqeql  5808  fmpt  5849  acexmidlemph  6068  inffiexmid  7203  ssfirab  7234  exmidaclem  7554  ioomax  10329  iccmax  10330  hashfibc  11261  dfphi2  12976  phiprmpw  12978  phisum  12997  unennn  13266  znnen  13267  isnsg4  13992  lssuni  14672  psrbagfi  14982  lgsquadlem2  16111