Theorem rabbidv 2802

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.)

Hypothesis

Ref	Expression
rabbidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
rabbidv	|- ( 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 rabbidv

Step	Hyp	Ref	Expression
1		rabbidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	adantr 276	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	rabbidva 2801	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   {crab 2524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-ral 2525  df-rab 2529

This theorem is referenced by:  rabeqbidv  2808  difeq2  3331  seex  4456  mptiniseg  5257  elovmporab  6254  supeq1  7277  supeq2  7280  supeq3  7281  cardcl  7477  isnumi  7478  cardval3ex  7481  carden2bex  7486  genpdflem  7822  genipv  7824  genpelxp  7826  addcomprg  7893  mulcomprg  7895  uzval  9855  ixxval  10229  fzval  10344  hashinfom  11141  hashennn  11143  ssenneg  11204  hashfibclem  11206  hashfibc  11207  shftfn  11509  bitsfval  12628  gcdval  12655  lcmval  12760  isprm  12806  odzval  12939  pceulem  12992  pceu  12993  pcval  12994  pczpre  12995  pcdiv  13000  lspval  14538  istopon  14878  toponsspwpwg  14887  clsval  14976  neival  15008  cnpval  15063  blvalps  15253  blval  15254  limccl  15524  ellimc3apf  15525  eldvap  15547  sgmval  15851  vtxdgfifival  16286  clwwlknon  16424  clwwlk0on0  16426  eupth2fi  16474