Theorem rabab 1822

Description: A class abstraction restricted to the universe is unrestricted.

Assertion

Ref	Expression
rabab	|- {x e. V | ph} = {x | ph}

Proof of Theorem rabab

Step	Hyp	Ref	Expression
1		df-rab 1652	. 2 |- {x e. V | ph} = {x | (x e. V /\ ph)}
2		pm3.27 323	. . . 4 |- ((x e. V /\ ph) -> ph)
3		visset 1813	. . . . 5 |- x e. V
4	3	jctl 290	. . . 4 |- (ph -> (x e. V /\ ph))
5	2, 4	impbi 157	. . 3 |- ((x e. V /\ ph) <-> ph)
6	5	abbii 1575	. 2 |- {x | (x e. V /\ ph)} = {x | ph}
7	1, 6	eqtr 1495	1 |- {x e. V | ph} = {x | ph}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  {crab 1648  Vcvv 1811

This theorem is referenced by:  elabs2 1964  intmin2 2557  iunab 2597  isumclimtf 7195  fctopOLD 7650  cctop 7652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812

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