Theorem rabbidv 2608

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 270	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	rabbidva 2607	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   e. wcel 1438   {crab 2363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-ral 2364  df-rab 2368

This theorem is referenced by:  rabeqbidv  2614  difeq2  3112  seex  4162  mptiniseg  4925  supeq1  6681  supeq2  6684  supeq3  6685  cardcl  6809  isnumi  6810  cardval3ex  6813  carden2bex  6817  genpdflem  7066  genipv  7068  genpelxp  7070  addcomprg  7137  mulcomprg  7139  uzval  9021  ixxval  9314  fzval  9426  hashinfom  10186  hashennn  10188  shftfn  10258  gcdval  11229  lcmval  11323  isprm  11369  istopon  11610  toponsspwpwg  11618