Theorem elabs 1964

Description: Membership in a class abstraction, expressed in terms of class substitution.

Hypothesis

Ref	Expression
elabs.1	|- A e. V

Assertion

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

Proof of Theorem elabs

Step	Hyp	Ref	Expression
1		elabs2 1962	. 2 |- (A e. {x | ph} <-> (A e. V /\ [A / x]ph))
2		elabs.1	. 2 |- A e. V
3	1, 2	mpbiran 727	1 |- (A e. {x | ph} <-> [A / x]ph)

Syntax hints:   <-> wb 146   e. wcel 957  [wsbc 1170  {cab 1463  Vcvv 1809

This theorem is referenced by:  intab 2557  hta 4715

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1651  df-v 1810  df-sbc 1940

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