Theorem elrab3 1906

Description: Membership in a restricted class abstraction with implicit substitution.

Hypothesis

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

Assertion

Ref	Expression
elrab3	|- (A e. B -> (A e. {x e. B | ph} <-> ps))

Distinct variable groups:   ps,x   x,A   x,B

Proof of Theorem elrab3

Step	Hyp	Ref	Expression
1		elrab.1	. . 3 |- (x = A -> (ph <-> ps))
2	1	elrab 1905	. 2 |- (A e. {x e. B | ph} <-> (A e. B /\ ps))
3	2	baib 685	1 |- (A e. B -> (A e. {x e. B | ph} <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {crab 1648

This theorem is referenced by:  unimax 2532

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-rab 1652  df-v 1812

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