Theorem rabbidv 2760

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 2759	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   e. wcel 2175   {crab 2487

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-ral 2488  df-rab 2492

This theorem is referenced by:  rabeqbidv  2766  difeq2  3284  seex  4381  mptiniseg  5176  elovmporab  6145  supeq1  7087  supeq2  7090  supeq3  7091  cardcl  7287  isnumi  7288  cardval3ex  7291  carden2bex  7296  genpdflem  7619  genipv  7621  genpelxp  7623  addcomprg  7690  mulcomprg  7692  uzval  9649  ixxval  10017  fzval  10131  hashinfom  10921  hashennn  10923  shftfn  11077  bitsfval  12195  gcdval  12222  lcmval  12327  isprm  12373  odzval  12506  pceulem  12559  pceu  12560  pcval  12561  pczpre  12562  pcdiv  12567  lspval  14094  istopon  14427  toponsspwpwg  14436  clsval  14525  neival  14557  cnpval  14612  blvalps  14802  blval  14803  limccl  15073  ellimc3apf  15074  eldvap  15096  sgmval  15397