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  4617  onsucmin  4656  seinxp  4848  fniniseg2  5834  fnniniseg2  5835  f1oresrab  5876  suppval1  6483  mptsuppd  6500  2omap  7324  2omapfi  7326  dfinfre  9294  hashfibclem  11304  minmax  12020  xrminmax  12056  iooinsup  12068  gcdass  12817  lcmass  12888  pcneg  13133  rrgsupp  14683  bdbl  15729  xmetxpbl  15734  lgsquadlem1  16401  lgsquadlem2  16402  2lgslem1a  16412  vtxdfifiun  16743  pw1map  17230