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  11106  bitsfval  12224  gcdval  12251  lcmval  12356  isprm  12402  odzval  12535  pceulem  12588  pceu  12589  pcval  12590  pczpre  12591  pcdiv  12596  lspval  14123  istopon  14456  toponsspwpwg  14465  clsval  14554  neival  14586  cnpval  14641  blvalps  14831  blval  14832  limccl  15102  ellimc3apf  15103  eldvap  15125  sgmval  15426