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  3271  seex  4366  mptiniseg  5160  elovmporab  6118  supeq1  7045  supeq2  7048  supeq3  7049  cardcl  7241  isnumi  7242  cardval3ex  7245  carden2bex  7249  genpdflem  7567  genipv  7569  genpelxp  7571  addcomprg  7638  mulcomprg  7640  uzval  9594  ixxval  9962  fzval  10076  hashinfom  10849  hashennn  10851  shftfn  10968  gcdval  12096  lcmval  12201  isprm  12247  odzval  12379  pceulem  12432  pceu  12433  pcval  12434  pczpre  12435  pcdiv  12440  lspval  13886  istopon  14181  toponsspwpwg  14190  clsval  14279  neival  14311  cnpval  14366  blvalps  14556  blval  14557  limccl  14813  ellimc3apf  14814  eldvap  14836