Theorem rabbidv 2749

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 2748	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   {crab 2476

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-ral 2477  df-rab 2481

This theorem is referenced by:  rabeqbidv  2755  difeq2  3272  seex  4367  mptiniseg  5161  elovmporab  6120  supeq1  7047  supeq2  7050  supeq3  7051  cardcl  7243  isnumi  7244  cardval3ex  7247  carden2bex  7251  genpdflem  7569  genipv  7571  genpelxp  7573  addcomprg  7640  mulcomprg  7642  uzval  9597  ixxval  9965  fzval  10079  hashinfom  10852  hashennn  10854  shftfn  10971  gcdval  12099  lcmval  12204  isprm  12250  odzval  12382  pceulem  12435  pceu  12436  pcval  12437  pczpre  12438  pcdiv  12443  lspval  13889  istopon  14192  toponsspwpwg  14201  clsval  14290  neival  14322  cnpval  14377  blvalps  14567  blval  14568  limccl  14838  ellimc3apf  14839  eldvap  14861