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  4426  mptiniseg  5223  elovmporab  6211  supeq1  7164  supeq2  7167  supeq3  7168  cardcl  7364  isnumi  7365  cardval3ex  7368  carden2bex  7373  genpdflem  7705  genipv  7707  genpelxp  7709  addcomprg  7776  mulcomprg  7778  uzval  9735  ixxval  10104  fzval  10218  hashinfom  11012  hashennn  11014  shftfn  11350  bitsfval  12468  gcdval  12495  lcmval  12600  isprm  12646  odzval  12779  pceulem  12832  pceu  12833  pcval  12834  pczpre  12835  pcdiv  12840  lspval  14369  istopon  14702  toponsspwpwg  14711  clsval  14800  neival  14832  cnpval  14887  blvalps  15077  blval  15078  limccl  15348  ellimc3apf  15349  eldvap  15371  sgmval  15672  vtxdgfifival  16050