Theorem rabbidva 2593

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)

Hypothesis

Ref	Expression
rabbidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rabbidva	|- ( ph -> { x e. A | ps } = { x e. A | ch } )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem rabbidva

Step	Hyp	Ref	Expression
1		rabbidva.1	. . 3 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	ralrimiva 2435	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
3		rabbi 2532	. 2 |- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )
4	2, 3	sylib 120	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   A.wral 2349   {crab 2353

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-ral 2354  df-rab 2358

This theorem is referenced by:  rabbidv  2594  rabeqbidva  2598  rabbi2dva  3175  rabxfrd  4221  onsucmin  4253  seinxp  4431  fniniseg2  5315  fnniniseg2  5316  f1oresrab  5355  dfinfre  8090  minmax  10239  gcdass  10537  lcmass  10600