Theorem rabbidv 2752

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 2751	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   {crab 2479

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-ral 2480  df-rab 2484

This theorem is referenced by:  rabeqbidv  2758  difeq2  3276  seex  4371  mptiniseg  5165  elovmporab  6127  supeq1  7061  supeq2  7064  supeq3  7065  cardcl  7261  isnumi  7262  cardval3ex  7265  carden2bex  7270  genpdflem  7593  genipv  7595  genpelxp  7597  addcomprg  7664  mulcomprg  7666  uzval  9622  ixxval  9990  fzval  10104  hashinfom  10889  hashennn  10891  shftfn  11008  bitsfval  12126  gcdval  12153  lcmval  12258  isprm  12304  odzval  12437  pceulem  12490  pceu  12491  pcval  12492  pczpre  12493  pcdiv  12498  lspval  14024  istopon  14357  toponsspwpwg  14366  clsval  14455  neival  14487  cnpval  14542  blvalps  14732  blval  14733  limccl  15003  ellimc3apf  15004  eldvap  15026  sgmval  15327