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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   {crab 2532

This proof depends on 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 proof 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 used by:  rabxmdc  3554  rabrsndc  3779  class2seteq  4302  dmmptg  5288  dmmptd  5517  fneqeql  5820  fmpt  5861  acexmidlemph  6081  inffiexmid  7217  ssfirab  7248  exmidaclem  7570  ioomax  10367  iccmax  10368  hashfibc  11305  dfphi2  13027  phiprmpw  13029  phisum  13048  unennn  13346  znnen  13347  isnsg4  14074  lssuni  14809  psrbagfi  15168  lgsquadlem2  16402  rabid1o  17239  stnot  17244