Theorem rabbidva 2803

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 2617	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
3		rabbi 2724	. 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 } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-ral 2527  df-rab 2531

This theorem is referenced by:  rabbidv  2804  rabeqbidva  2811  rabbi2dva  3433  rabxfrd  4595  onsucmin  4634  seinxp  4826  fniniseg2  5805  fnniniseg2  5806  f1oresrab  5847  suppval1  6452  mptsuppd  6469  2omap  7282  2omapfi  7284  dfinfre  9250  hashfibclem  11234  minmax  11944  xrminmax  11979  iooinsup  11991  gcdass  12740  lcmass  12811  pcneg  13052  rrgsupp  14516  bdbl  15498  xmetxpbl  15503  lgsquadlem1  16080  lgsquadlem2  16081  2lgslem1a  16091  vtxdfifiun  16422  pw1map  16909