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  4380  mptiniseg  5174  elovmporab  6136  supeq1  7070  supeq2  7073  supeq3  7074  cardcl  7270  isnumi  7271  cardval3ex  7274  carden2bex  7279  genpdflem  7602  genipv  7604  genpelxp  7606  addcomprg  7673  mulcomprg  7675  uzval  9632  ixxval  10000  fzval  10114  hashinfom  10904  hashennn  10906  shftfn  11054  bitsfval  12172  gcdval  12199  lcmval  12304  isprm  12350  odzval  12483  pceulem  12536  pceu  12537  pcval  12538  pczpre  12539  pcdiv  12544  lspval  14070  istopon  14403  toponsspwpwg  14412  clsval  14501  neival  14533  cnpval  14588  blvalps  14778  blval  14779  limccl  15049  ellimc3apf  15050  eldvap  15072  sgmval  15373