Theorem rabbidva 2607

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (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 2446	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
3		rabbi 2544	. 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 1289   e. wcel 1438   A.wral 2359   {crab 2363

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-ral 2364  df-rab 2368

This theorem is referenced by:  rabbidv  2608  rabeqbidva  2615  rabbi2dva  3208  rabxfrd  4291  onsucmin  4324  seinxp  4509  fniniseg2  5421  fnniniseg2  5422  f1oresrab  5463  dfinfre  8417  minmax  10661  gcdass  11282  lcmass  11345