Theorem rabbidv 2792

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 2791	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2202   {crab 2515

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-ral 2516  df-rab 2520

This theorem is referenced by:  rabeqbidv  2798  difeq2  3321  seex  4438  mptiniseg  5238  elovmporab  6232  supeq1  7228  supeq2  7231  supeq3  7232  cardcl  7428  isnumi  7429  cardval3ex  7432  carden2bex  7437  genpdflem  7770  genipv  7772  genpelxp  7774  addcomprg  7841  mulcomprg  7843  uzval  9801  ixxval  10175  fzval  10290  hashinfom  11086  hashennn  11088  shftfn  11447  bitsfval  12566  gcdval  12593  lcmval  12698  isprm  12744  odzval  12877  pceulem  12930  pceu  12931  pcval  12932  pczpre  12933  pcdiv  12938  lspval  14469  istopon  14807  toponsspwpwg  14816  clsval  14905  neival  14937  cnpval  14992  blvalps  15182  blval  15183  limccl  15453  ellimc3apf  15454  eldvap  15476  sgmval  15780  vtxdgfifival  16215  clwwlknon  16353  clwwlk0on0  16355  eupth2fi  16403