Theorem rabeqbidv 2290

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

Proof of Theorem rabeqbidv

Step	Hyp	Ref	Expression
1		rabeqbidv.1	. . 3 |- (ph -> A = B)
2		rabeq 2289	. . 3 |- (A = B -> {x e. A | ps} = {x e. B | ps})
3	1, 2	syl 12	. 2 |- (ph -> {x e. A | ps} = {x e. B | ps})
4		rabeqbidv.2	. . 3 |- (ph -> (ps <-> ch))
5	4	rabbidv 2287	. 2 |- (ph -> {x e. B | ps} = {x e. B | ch})
6	3, 5	eqtrd 1925	1 |- (ph -> {x e. A | ps} = {x e. B | ch})

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298  {crab 2108

This theorem is referenced by:  ubos2 14598  mxlelt2 14606  mxlelt 14607  mnlelt2 14608  mnlmxl2 14611  rngisoval 16131  idlval 16161  pridlval 16181  maxidlval 16187  patoms 17000  plusssfval 17204  ocvfval 17206  lineset 17219  pmapfval 17236  paddfval 17258  dilfset 17401  trnfset 17404  trnset 17405

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112

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