Theorem rabeqbidv 2758

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 2755	. . 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 2752	. 2 |- ( ph -> { x e. B | ps } = { x e. B | ch } )
6	3, 5	eqtrd 2229	1 |- ( ph -> { x e. A | ps } = { x e. B | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   {crab 2479

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484

This theorem is referenced by:  elfvmptrab1  5659  elovmporab1w  6128  mpoxopoveq  6307  supeq123d  7066  phival  12408  dfphi2  12415  gsumress  13099  ismhm  13165  mhmex  13166  issubm  13176  issubg  13381  subgex  13384  isnsg  13410  dfrhm2  13788  isrim0  13795  issubrng  13833  issubrg  13855  rrgval  13896  lsssetm  13990  mplvalcoe  14324  cldval  14443  neifval  14484  cnfval  14538  cnpfval  14539  cnprcl2k  14550  hmeofvalg  14647  ispsmet  14667  ismet  14688  isxmet  14689  blfvalps  14729  cncfval  14916