Theorem eq2ab 1549

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 1443	. . 3 |- (y e. {x | ph} -> A.x y e. {x | ph})
2		hbab1 1443	. . 3 |- (y e. {x | ps} -> A.x y e. {x | ps})
3	1, 2	cleqf 1536	. 2 |- ({x | ph} = {x | ps} <-> A.x(x e. {x | ph} <-> x e. {x | ps}))
4		abid 1442	. . . 4 |- (x e. {x | ph} <-> ph)
5		abid 1442	. . . 4 |- (x e. {x | ps} <-> ps)
6	4, 5	bibi12i 608	. . 3 |- ((x e. {x | ph} <-> x e. {x | ps}) <-> (ph <-> ps))
7	6	albii 975	. 2 |- (A.x(x e. {x | ph} <-> x e. {x | ps}) <-> A.x(ph <-> ps))
8	3, 7	bitr 173	1 |- ({x | ph} = {x | ps} <-> A.x(ph <-> ps))

Syntax hints:   <-> wb 146  A.wal 950   = wceq 1099   e. wcel 1105  {cab 1440

This theorem is referenced by:  abbii 1551  abbid 1552  pw2en 4380  karden 4650

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449

Copyright terms: Public domain