Theorem rabbidva 2719

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 2544	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
3		rabbi 2648	. 2 |- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )
4	2, 3	sylib 121	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1349   e. wcel 2142   A.wral 2449   {crab 2453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-11 1500  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-ral 2454  df-rab 2458

This theorem is referenced by:  rabbidv  2720  rabeqbidva  2727  rabbi2dva  3336  rabxfrd  4455  onsucmin  4492  seinxp  4683  fniniseg2  5622  fnniniseg2  5623  f1oresrab  5665  dfinfre  8876  minmax  11197  xrminmax  11232  iooinsup  11244  gcdass  11974  lcmass  12043  pcneg  12282  bdbl  13382  xmetxpbl  13387