Theorem rabeqbidv 2816

Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)

Hypotheses

Ref	Expression
rabeqbidv.1	|- ( ph -> A = B )
rabeqbidv.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
rabeqbidv	|- ( ph -> { x e. A | ps } = { x e. B | ch } )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   ps( x)   ch( x)

Proof of Theorem rabeqbidv

Step	Hyp	Ref	Expression
1		rabeqbidv.1	. . 3 |- ( ph -> A = B )
2		rabeq 2813	. . 3 |- ( A = B -> { x e. A | ps } = { x e. B | ps } )
3	1, 2	syl 14	. 2 |- ( ph -> { x e. A | ps } = { x e. B | ps } )
4		rabeqbidv.2	. . 3 |- ( ph -> ( ps <-> ch ) )
5	4	rabbidv 2810	. 2 |- ( ph -> { x e. B | ps } = { x e. B | ch } )
6	3, 5	eqtrd 2271	1 |- ( ph -> { x e. A | ps } = { x e. B | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   {crab 2532

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537

This theorem is referenced by:  elfvmptrab1  5794  elovmporab1w  6280  suppval  6467  mpoxopoveq  6501  supeq123d  7321  phival  12969  dfphi2  12976  gzsumress  13689  ismhm  13745  mhmex  13746  issubm  13756  issubg  13953  subgex  13956  isnsg  13982  dfrhm2  14434  isrim0  14441  issubrng  14480  issubrg  14502  rrgval  14543  lsssetm  14665  mplvalcoe  15004  cldval  15123  neifval  15164  cnfval  15218  cnpfval  15219  cnprcl2k  15230  hmeofvalg  15327  ispsmet  15347  ismet  15368  isxmet  15369  blfvalps  15409  cncfval  15596  vtxdgfval  16443  vtxdgop  16447  vtxdeqd  16451  clwwlkg  16548  clwwlkng  16560