Theorem rabid2 2654

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 2286	. . 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 1470	. . 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 2464	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eqeq2i 2188	. 2 |- ( A = { x e. A | ph } <-> A = { x | ( x e. A /\ ph ) } )
7		df-ral 2460	. 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 1351   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   {crab 2459

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-rab 2464

This theorem is referenced by:  rabxmdc  3455  rabrsndc  3661  class2seteq  4164  dmmptg  5127  dmmptd  5347  fneqeql  5625  fmpt  5667  acexmidlemph  5868  inffiexmid  6906  ssfirab  6933  exmidaclem  7207  ioomax  9948  iccmax  9949  dfphi2  12220  phiprmpw  12222  phisum  12240  unennn  12398  znnen  12399  isnsg4  13072