Theorem elab3 1894

Description: Membership in a class abstraction using implicit substitution.

Hypotheses

Ref	Expression
elab3.1	|- (ps -> A e. V)
elab3.2	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
elab3	|- (A e. {x | ph} <-> ps)

Distinct variable groups:   ps,x   x,A

Proof of Theorem elab3

Step	Hyp	Ref	Expression
1		elab3.1	. 2 |- (ps -> A e. V)
2		elab3.2	. . 3 |- (x = A -> (ph <-> ps))
3	2	elab3g 1893	. 2 |- ((ps -> A e. V) -> (A e. {x | ph} <-> ps))
4	1, 3	ax-mp 7	1 |- (A e. {x | ph} <-> ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802

This theorem is referenced by:  fvelrnb 3745  oprvalelrn 4024  elpm 4320  elq 6195  eltg3t 7568  islp 7685  islno 8348

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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