Theorem rabbidva 2713

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 2538	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
3		rabbi 2642	. 2 |- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )
4	2, 3	sylib 121	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2443   {crab 2447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-ral 2448  df-rab 2452

This theorem is referenced by:  rabbidv  2714  rabeqbidva  2721  rabbi2dva  3329  rabxfrd  4446  onsucmin  4483  seinxp  4674  fniniseg2  5606  fnniniseg2  5607  f1oresrab  5649  dfinfre  8847  minmax  11167  xrminmax  11202  iooinsup  11214  gcdass  11944  lcmass  12013  pcneg  12252  bdbl  13103  xmetxpbl  13108