Theorem elab3 1903

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 1902	. 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 956   e. wcel 958  {cab 1463  Vcvv 1811

This theorem is referenced by:  fvelrnb 3760  oprvalelrn 4039  elpm 4336  isfi 4382  elq 6257  eltg3t 7626  islp 7744  islno 8414

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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