Theorem elabg 1899

Description: Membership in a class abstraction with implicit substitution. Compare Theorem 6.13 of [Quine] p. 44.

Hypothesis

Ref	Expression
elabg.1	|- (x = A -> (ph <-> ps))

Assertion

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

Distinct variable groups:   ps,x   x,A

Proof of Theorem elabg

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (y e. A -> A.x y e. A)
2		ax-17 971	. 2 |- (ps -> A.xps)
3		elabg.1	. 2 |- (x = A -> (ph <-> ps))
4	1, 2, 3	elabgf 1898	1 |- (A e. B -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {cab 1463

This theorem is referenced by:  elab2g 1900  elab3g 1902  intmin3 2558  finds 3156  scott0 4717  elcncf 7265  eltgt 7618  eltg2t 7619  iscld 7669  dfpjopt 10111  spfi 10445  spfiOLD 10446  ishomeo 10517  rcfpfillem1 10585  rcfpfillem1OLD 10586  eloi 10659  ismonb 10738  isepib 10748  isfunb 10755

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-v 1812

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