Theorem rabid2 2254

Description: An "identity" law for restricted class abstraction. (The proof was 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

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		abeq2 1999	. . 3 |- (A = {x | (x e. A /\ ph)} <-> A.x(x e. A <-> (x e. A /\ ph)))
2		pm4.71 697	. . . 4 |- ((x e. A -> ph) <-> (x e. A <-> (x e. A /\ ph)))
3	2	albii 1346	. . 3 |- (A.x(x e. A -> ph) <-> A.x(x e. A <-> (x e. A /\ ph)))
4	1, 3	bitr4i 193	. 2 |- (A = {x | (x e. A /\ ph)} <-> A.x(x e. A -> ph))
5		df-rab 2112	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
6	5	eqeq2i 1894	. 2 |- (A = {x e. A | ph} <-> A = {x | (x e. A /\ ph)})
7		df-ral 2109	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
8	4, 6, 7	3bitr4i 200	1 |- (A = {x e. A | ph} <-> A.x e. A ph)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  {crab 2108

This theorem is referenced by:  iinrab2 3319  class2seteq 3472  zfrep6 4545  abrexex 4836  ioomax 7561  tfisg 13912  wfisg 13917  frinsg 13941  rabxm 15667  fdc 15812  pmap1 17247  pol1 17323

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rab 2112

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