Theorem rabbidv 2788

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 2787	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-ral 2513  df-rab 2517

This theorem is referenced by:  rabeqbidv  2794  difeq2  3316  seex  4425  mptiniseg  5222  elovmporab  6204  supeq1  7149  supeq2  7152  supeq3  7153  cardcl  7349  isnumi  7350  cardval3ex  7353  carden2bex  7358  genpdflem  7690  genipv  7692  genpelxp  7694  addcomprg  7761  mulcomprg  7763  uzval  9720  ixxval  10088  fzval  10202  hashinfom  10995  hashennn  10997  shftfn  11330  bitsfval  12448  gcdval  12475  lcmval  12580  isprm  12626  odzval  12759  pceulem  12812  pceu  12813  pcval  12814  pczpre  12815  pcdiv  12820  lspval  14348  istopon  14681  toponsspwpwg  14690  clsval  14779  neival  14811  cnpval  14866  blvalps  15056  blval  15057  limccl  15327  ellimc3apf  15328  eldvap  15350  sgmval  15651