Theorem elrab3 1909

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 1908	. 2 |- (A e. {x e. B | ph} <-> (A e. B /\ ps))
3	2	baib 687	1 |- (A e. B -> (A e. {x e. B | ph} <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  {crab 1651

This theorem is referenced by:  unimax 2536

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655  df-v 1815

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