Theorem rabbidva 2809

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.)

Hypothesis

Ref	Expression
rabbidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rabbidva	|- ( 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 rabbidva

Step	Hyp	Ref	Expression
1		rabbidva.1	. . 3 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	ralrimiva 2623	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
3		rabbi 2730	. 2 |- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )
4	2, 3	sylib 122	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-ral 2533  df-rab 2537

This theorem is used by:  rabbidv  2810  rabeqbidva  2817  rabbi2dva  3439  rabxfrd  4615  onsucmin  4654  seinxp  4846  fniniseg2  5831  fnniniseg2  5832  f1oresrab  5873  suppval1  6479  mptsuppd  6496  2omap  7319  2omapfi  7321  dfinfre  9289  hashfibclem  11298  minmax  12014  xrminmax  12050  iooinsup  12062  gcdass  12811  lcmass  12882  pcneg  13127  rrgsupp  14658  bdbl  15695  xmetxpbl  15700  lgsquadlem1  16362  lgsquadlem2  16363  2lgslem1a  16373  vtxdfifiun  16704  pw1map  17191