Theorem eq2ab 2004

Description: Equality of two class abstractions means their wff's are equivalent.

Assertion

Ref	Expression
eq2ab	|- ({x | ph} = {x | ps} <-> A.x(ph <-> ps))

Proof of Theorem eq2ab

Step	Hyp	Ref	Expression
1		hbab1 1874	. . 3 |- (y e. {x | ph} -> A.x y e. {x | ph})
2		hbab1 1874	. . 3 |- (y e. {x | ps} -> A.x y e. {x | ps})
3	1, 2	cleqf 1984	. 2 |- ({x | ph} = {x | ps} <-> A.x(x e. {x | ph} <-> x e. {x | ps}))
4		abid 1873	. . . 4 |- (x e. {x | ph} <-> ph)
5		abid 1873	. . . 4 |- (x e. {x | ps} <-> ps)
6	4, 5	bibi12i 672	. . 3 |- ((x e. {x | ph} <-> x e. {x | ps}) <-> (ph <-> ps))
7	6	albii 1346	. 2 |- (A.x(x e. {x | ph} <-> x e. {x | ps}) <-> A.x(ph <-> ps))
8	3, 7	bitri 190	1 |- ({x | ph} = {x | ps} <-> A.x(ph <-> ps))

Syntax hints:   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871

This theorem is referenced by:  abbii 2006  abbid 2007  dfiota2 5090  iotabi 5094  uniabio 5095  iotanul 5098  pw2en 5505  karden 5856  elnev 16404

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-10 1308  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-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880

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