Theorem elabs2 1967

Description: Membership in a class abstraction, expressed in terms of class substitution. Theorem 6.13 of [Quine] p. 44.

Assertion

Ref	Expression
elabs2	|- (A e. {x | ph} <-> (A e. V /\ [A / x]ph))

Proof of Theorem elabs2

Step	Hyp	Ref	Expression
1		rabab 1825	. . 3 |- {x e. V | ph} = {x | ph}
2	1	eleq2i 1541	. 2 |- (A e. {x e. V | ph} <-> A e. {x | ph})
3		ax-17 973	. . 3 |- (y e. V -> A.x y e. V)
4	3	elrabsf 1966	. 2 |- (A e. {x e. V | ph} <-> (A e. V /\ [A / x]ph))
5	2, 4	bitr3 175	1 |- (A e. {x | ph} <-> (A e. V /\ [A / x]ph))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 960  [wsbc 1172  {cab 1466  {crab 1651  Vcvv 1814

This theorem is referenced by:  elabsg 1968  elabs 1969

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655  df-v 1815  df-sbc 1945

Copyright terms: Public domain