Theorem rabbidva 2791

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 2606	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
3		rabbi 2712	. 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 2202   A.wral 2511   {crab 2515

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-ral 2516  df-rab 2520

This theorem is referenced by:  rabbidv  2792  rabeqbidva  2799  rabbi2dva  3417  rabxfrd  4572  onsucmin  4611  seinxp  4803  fniniseg2  5778  fnniniseg2  5779  f1oresrab  5820  suppval1  6417  mptsuppd  6434  dfinfre  9195  minmax  11870  xrminmax  11905  iooinsup  11917  gcdass  12666  lcmass  12737  pcneg  12978  rrgsupp  14361  bdbl  15314  xmetxpbl  15319  lgsquadlem1  15896  lgsquadlem2  15897  2lgslem1a  15907  vtxdfifiun  16238  2omap  16715  pw1map  16717