Theorem rabeq12OLD 15665

Description: Equality of restricted class abstractions. (Moved to rabeqbidv 2290 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Sep-2011.)

Hypotheses

Ref	Expression
rabeq12.1OLD	|- (ph -> A = B)
rabeq12.2OLD	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
rabeq12OLD	|- (ph -> {x e. A | ps} = {x e. B | ch})

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

Proof of Theorem rabeq12OLD

Step	Hyp	Ref	Expression
1		rabeq12.1OLD	. . 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		rabeq12.2OLD	. . 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 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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