Theorem rabbidva 2764

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 2581	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
3		rabbi 2686	. 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 1373   e. wcel 2178   A.wral 2486   {crab 2490

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-ral 2491  df-rab 2495

This theorem is referenced by:  rabbidv  2765  rabeqbidva  2772  rabbi2dva  3389  rabxfrd  4534  onsucmin  4573  seinxp  4764  fniniseg2  5725  fnniniseg2  5726  f1oresrab  5768  dfinfre  9064  minmax  11656  xrminmax  11691  iooinsup  11703  gcdass  12451  lcmass  12522  pcneg  12763  bdbl  15090  xmetxpbl  15095  lgsquadlem1  15669  lgsquadlem2  15670  2lgslem1a  15680  2omap  16132  pw1map  16134